Affine curves - Plane and Space Curves

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# Affine curves

Affine curves in Sage are curves in an affine space or an affine plane.

EXAMPLES:

We can construct curves in either an affine plane:

Sage

    sage: A.<x,y> = AffineSpace(QQ, 2)
    sage: C = Curve([y - x^2], A); C
    Affine Plane Curve over Rational Field defined by -x^2 + y

Python

    >>> from sage.all import *
    >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
    >>> C = Curve([y - x**Integer(2)], A); C
    Affine Plane Curve over Rational Field defined by -x^2 + y

or in higher dimensional affine space:

Sage

    sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
    sage: C = Curve([y - x^2, z - w^3, w - y^4], A); C
    Affine Curve over Rational Field defined by -x^2 + y, -w^3 + z, -y^4 + w

[Python]

    >>> from sage.all import *
    >>> A = AffineSpace(QQ, Integer(4), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = A._first_ngens(4)
    >>> C = Curve([y - x**Integer(2), z - w**Integer(3), w - y**Integer(4)], A); C
    Affine Curve over Rational Field defined by -x^2 + y, -w^3 + z, -y^4 + w

## Integral affine curves over finite fields

If the curve is defined over a finite field and integral, that is reduced and irreducible, its function field is tightly coupled with the curve so that advanced computations based on Sage’s global function field machinery are available.

EXAMPLES:

Sage

    sage: k.<a> = GF(2)
    sage: A.<x,y,z> = AffineSpace(k, 3)
    sage: C = Curve([x^2 + x - y^3, y^4 - y - z^3], A)
    sage: C.genus()
    10
    sage: C.function_field()
    Function field in z defined by z^9 + x^8 + x^6 + x^5 + x^4 + x^3 + x

Python

    >>> from sage.all import *
    >>> k = GF(Integer(2), names=('a',)); (a,) = k._first_ngens(1)
    >>> A = AffineSpace(k, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
    >>> C = Curve([x**Integer(2) + x - y**Integer(3), y**Integer(4) - y - z**Integer(3)], A)
    >>> C.genus()
    10
    >>> C.function_field()
    Function field in z defined by z^9 + x^8 + x^6 + x^5 + x^4 + x^3 + x

Closed points of arbitrary degree can be computed:

Sage

    sage: # long time
    sage: C.closed_points()
    [Point (x, y, z), Point (x + 1, y, z)]
    sage: C.closed_points(2)
    [Point (x^2 + x + 1, y + 1, z),
     Point (y^2 + y + 1, x + y, z),
     Point (y^2 + y + 1, x + y + 1, z)]
    sage: p = _[0]
    sage: p.places()
    [Place (x^2 + x + 1, (1/(x^4 + x^2 + 1))*z^7 + (1/(x^4 + x^2 + 1))*z^6 + 1)]

[Python]

    >>> from sage.all import *
    >>> # long time
    >>> C.closed_points()
    [Point (x, y, z), Point (x + 1, y, z)]
    >>> C.closed_points(Integer(2))
    [Point (x^2 + x + 1, y + 1, z),
     Point (y^2 + y + 1, x + y, z),
     Point (y^2 + y + 1, x + y + 1, z)]
    >>> p = _[Integer(0)]
    >>> p.places()
    [Place (x^2 + x + 1, (1/(x^4 + x^2 + 1))*z^7 + (1/(x^4 + x^2 + 1))*z^6 + 1)]

The places at infinity correspond to the extra closed points of the curve’s projective closure:

Sage

    sage: C.places_at_infinity()                # long time
    [Place (1/x, 1/x*z)]

Python

    >>> from sage.all import *
    >>> C.places_at_infinity()                # long time
    [Place (1/x, 1/x*z)]

It is easy to transit to and from the function field of the curve:

Sage

    sage: fx = C(x)
    sage: fy = C(y)
    sage: fx^2 + fx - fy^3
    0
    sage: fx.divisor()
    -9*Place (1/x, 1/x*z)
     + 9*Place (x, z)
    sage: p, = fx.zeros()
    sage: C.place_to_closed_point(p)
    Point (x, y, z)
    sage: _.rational_point()
    (0, 0, 0)
    sage: _.closed_point()
    Point (x, y, z)
    sage: _.place()
    Place (x, z)

[Python]

    >>> from sage.all import *
    >>> fx = C(x)
    >>> fy = C(y)
    >>> fx**Integer(2) + fx - fy**Integer(3)
    0
    >>> fx.divisor()
    -9*Place (1/x, 1/x*z)
     + 9*Place (x, z)
    >>> p, = fx.zeros()
    >>> C.place_to_closed_point(p)
    Point (x, y, z)
    >>> _.rational_point()
    (0, 0, 0)
    >>> _.closed_point()
    Point (x, y, z)
    >>> _.place()
    Place (x, z)

## Integral affine curves over \(\QQ\)

An integral curve over \(\QQ\) is equipped also with the function field. Unlike over finite fields, it is not possible to enumerate closed points.

EXAMPLES:

Sage

    sage: A.<x,y> = AffineSpace(QQ, 2)
    sage: C = Curve(x^2 + y^2 -1)
    sage: p = C(0,1)
    sage: p
    (0, 1)
    sage: p.closed_point()
    Point (x, y - 1)
    sage: pl = _.place()
    sage: C.parametric_representation(pl)
    (s + ..., 1 - 1/2*s^2 - 1/8*s^4 - 1/16*s^6 + ...)
    sage: sx, sy = _
    sage: sx = sx.polynomial(10); sx
    s
    sage: sy = sy.polynomial(10); sy
    -7/256*s^10 - 5/128*s^8 - 1/16*s^6 - 1/8*s^4 - 1/2*s^2 + 1
    sage: s = var('s')                                                                  # needs sage.symbolic
    sage: P1 = parametric_plot([sx, sy], (s, -1, 1), color='red')                       # needs sage.plot sage.symbolic
    sage: P2 = C.plot((x, -1, 1), (y, 0, 2))  # half circle                             # needs sage.plot sage.symbolic
    sage: P1 + P2                                                                       # needs sage.plot sage.symbolic
    Graphics object consisting of 2 graphics primitives

Python

    >>> from sage.all import *
    >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
    >>> C = Curve(x**Integer(2) + y**Integer(2) -Integer(1))
    >>> p = C(Integer(0),Integer(1))
    >>> p
    (0, 1)
    >>> p.closed_point()
    Point (x, y - 1)
    >>> pl = _.place()
    >>> C.parametric_representation(pl)
    (s + ..., 1 - 1/2*s^2 - 1/8*s^4 - 1/16*s^6 + ...)
    >>> sx, sy = _
    >>> sx = sx.polynomial(Integer(10)); sx
    s
    >>> sy = sy.polynomial(Integer(10)); sy
    -7/256*s^10 - 5/128*s^8 - 1/16*s^6 - 1/8*s^4 - 1/2*s^2 + 1
    >>> s = var('s')                                                                  # needs sage.symbolic
    >>> P1 = parametric_plot([sx, sy], (s, -Integer(1), Integer(1)), color='red')                       # needs sage.plot sage.symbolic
    >>> P2 = C.plot((x, -Integer(1), Integer(1)), (y, Integer(0), Integer(2)))  # half circle                             # needs sage.plot sage.symbolic
    >>> P1 + P2                                                                       # needs sage.plot sage.symbolic
    Graphics object consisting of 2 graphics primitives

AUTHORS:

- William Stein (2005-11-13)

- David Joyner (2005-11-13)

- David Kohel (2006-01)

- Grayson Jorgenson (2016-08)

- Kwankyu Lee (2019-05): added integral affine curves

class sage.schemes.curves.affine_curve.AffineCurve(A, X)

  Bases: `Curve_generic`, `AlgebraicScheme_subscheme_affine`

  Affine curves.

  EXAMPLES:

  Sage

      sage: # needs sage.rings.number_field
      sage: R.<v> = QQ[]
      sage: K.<u> = NumberField(v^2 + 3)
      sage: A.<x,y,z> = AffineSpace(K, 3)
      sage: C = Curve([z - u*x^2, y^2], A); C
      Affine Curve over Number Field in u with defining polynomial v^2 + 3
      defined by (-u)*x^2 + z, y^2

  Python

      >>> from sage.all import *
      >>> # needs sage.rings.number_field
      >>> R = QQ['v']; (v,) = R._first_ngens(1)
      >>> K = NumberField(v**Integer(2) + Integer(3), names=('u',)); (u,) = K._first_ngens(1)
      >>> A = AffineSpace(K, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
      >>> C = Curve([z - u*x**Integer(2), y**Integer(2)], A); C
      Affine Curve over Number Field in u with defining polynomial v^2 + 3
      defined by (-u)*x^2 + z, y^2

  Sage

      sage: A.<x,y,z> = AffineSpace(GF(7), 3)
      sage: C = Curve([x^2 - z, z - 8*x], A); C
      Affine Curve over Finite Field of size 7 defined by x^2 - z, -x + z

  [Python]

      >>> from sage.all import *
      >>> A = AffineSpace(GF(Integer(7)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
      >>> C = Curve([x**Integer(2) - z, z - Integer(8)*x], A); C
      Affine Curve over Finite Field of size 7 defined by x^2 - z, -x + z

  projective_closure(i=0, PP=None)

    Return the projective closure of this affine curve.

    INPUT:

    - `i` – (default: 0) the index of the affine coordinate chart of the projective space that the affine ambient space of this curve embeds into

    - `PP` – (default: `None`) ambient projective space to compute the projective closure in; this is constructed if it is not given

    OUTPUT: a curve in projective space

    EXAMPLES:

    Sage

        sage: A.<x,y,z> = AffineSpace(QQ, 3)
        sage: C = Curve([y-x^2,z-x^3], A)
        sage: C.projective_closure()
        Projective Curve over Rational Field defined by x1^2 - x0*x2,
        x1*x2 - x0*x3, x2^2 - x1*x3

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
        >>> C = Curve([y-x**Integer(2),z-x**Integer(3)], A)
        >>> C.projective_closure()
        Projective Curve over Rational Field defined by x1^2 - x0*x2,
        x1*x2 - x0*x3, x2^2 - x1*x3

    Sage

        sage: A.<x,y,z> = AffineSpace(QQ, 3)
        sage: C = Curve([y - x^2, z - x^3], A)
        sage: C.projective_closure()
        Projective Curve over Rational Field defined by
        x1^2 - x0*x2, x1*x2 - x0*x3, x2^2 - x1*x3

    [Python]

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
        >>> C = Curve([y - x**Integer(2), z - x**Integer(3)], A)
        >>> C.projective_closure()
        Projective Curve over Rational Field defined by
        x1^2 - x0*x2, x1*x2 - x0*x3, x2^2 - x1*x3

    Sage

        sage: A.<x,y> = AffineSpace(CC, 2)
        sage: C = Curve(y - x^3 + x - 1, A)
        sage: C.projective_closure(1)
        Projective Plane Curve over Complex Field with 53 bits of precision defined by
        x0^3 - x0*x1^2 + x1^3 - x1^2*x2

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(CC, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(y - x**Integer(3) + x - Integer(1), A)
        >>> C.projective_closure(Integer(1))
        Projective Plane Curve over Complex Field with 53 bits of precision defined by
        x0^3 - x0*x1^2 + x1^3 - x1^2*x2

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: P.<u,v,w> = ProjectiveSpace(QQ, 2)
        sage: C = Curve([y - x^2], A)
        sage: C.projective_closure(1, P).ambient_space() == P
        True

    [Python]

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> P = ProjectiveSpace(QQ, Integer(2), names=('u', 'v', 'w',)); (u, v, w,) = P._first_ngens(3)
        >>> C = Curve([y - x**Integer(2)], A)
        >>> C.projective_closure(Integer(1), P).ambient_space() == P
        True

class sage.schemes.curves.affine_curve.AffineCurve_field(A, X)

  Bases: `AffineCurve`, `AlgebraicScheme_subscheme_affine_field`

  Affine curves over fields.

  blowup(P=None)

    Return the blow up of this affine curve at the point `P`.

    The blow up is described by affine charts. This curve must be irreducible.

    INPUT:

    - `P` – (default: `None`) a point on this curve at which to blow up; if `None`, then `P` is taken to be the origin

    OUTPUT: a tuple of

    - a tuple of curves in affine space of the same dimension as the ambient space of this curve, which define the blow up in each affine chart.

    - a tuple of tuples such that the j-th element of the i-th tuple is the transition map from the i-th affine patch to the j-th affine patch.

    - a tuple consisting of the restrictions of the projection map from the blow up back to the original curve, restricted to each affine patch. There the i-th element will be the projection from the i-th affine patch.

    EXAMPLES:

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve([y^2 - x^3], A)
        sage: C.blowup()
        ((Affine Plane Curve over Rational Field defined by s1^2 - x,
          Affine Plane Curve over Rational Field defined by y*s0^3 - 1),
        ([Scheme endomorphism of Affine Plane Curve over Rational Field
           defined by s1^2 - x
            Defn: Defined on coordinates by sending (x, s1) to (x, s1),
          Scheme morphism:
            From: Affine Plane Curve over Rational Field defined by s1^2 - x
            To:   Affine Plane Curve over Rational Field defined by y*s0^3 - 1
            Defn: Defined on coordinates by sending (x, s1) to (x*s1, 1/s1)],
         [Scheme morphism:
            From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1
            To:   Affine Plane Curve over Rational Field defined by s1^2 - x
            Defn: Defined on coordinates by sending (y, s0) to (y*s0, 1/s0),
          Scheme endomorphism of Affine Plane Curve over Rational Field
           defined by y*s0^3 - 1
            Defn: Defined on coordinates by sending (y, s0) to (y, s0)]),
        (Scheme morphism:
           From: Affine Plane Curve over Rational Field defined by s1^2 - x
           To:   Affine Plane Curve over Rational Field defined by -x^3 + y^2
           Defn: Defined on coordinates by sending (x, s1) to (x, x*s1),
         Scheme morphism:
           From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1
           To:   Affine Plane Curve over Rational Field defined by -x^3 + y^2
           Defn: Defined on coordinates by sending (y, s0) to (y*s0, y)))

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([y**Integer(2) - x**Integer(3)], A)
        >>> C.blowup()
        ((Affine Plane Curve over Rational Field defined by s1^2 - x,
          Affine Plane Curve over Rational Field defined by y*s0^3 - 1),
        ([Scheme endomorphism of Affine Plane Curve over Rational Field
           defined by s1^2 - x
            Defn: Defined on coordinates by sending (x, s1) to (x, s1),
          Scheme morphism:
            From: Affine Plane Curve over Rational Field defined by s1^2 - x
            To:   Affine Plane Curve over Rational Field defined by y*s0^3 - 1
            Defn: Defined on coordinates by sending (x, s1) to (x*s1, 1/s1)],
         [Scheme morphism:
            From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1
            To:   Affine Plane Curve over Rational Field defined by s1^2 - x
            Defn: Defined on coordinates by sending (y, s0) to (y*s0, 1/s0),
          Scheme endomorphism of Affine Plane Curve over Rational Field
           defined by y*s0^3 - 1
            Defn: Defined on coordinates by sending (y, s0) to (y, s0)]),
        (Scheme morphism:
           From: Affine Plane Curve over Rational Field defined by s1^2 - x
           To:   Affine Plane Curve over Rational Field defined by -x^3 + y^2
           Defn: Defined on coordinates by sending (x, s1) to (x, x*s1),
         Scheme morphism:
           From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1
           To:   Affine Plane Curve over Rational Field defined by -x^3 + y^2
           Defn: Defined on coordinates by sending (y, s0) to (y*s0, y)))

    Sage

        sage: # needs sage.rings.number_field
        sage: K.<a> = QuadraticField(2)
        sage: A.<x,y,z> = AffineSpace(K, 3)
        sage: C = Curve([y^2 - a*x^5, x - z], A)
        sage: B = C.blowup()
        sage: B[0]
        (Affine Curve over Number Field in a with defining polynomial x^2 - 2
          with a = 1.414213562373095? defined by s2 - 1, 2*x^3 + (-a)*s1^2,
         Affine Curve over Number Field in a with defining polynomial x^2 - 2
          with a = 1.414213562373095? defined by s0 - s2, 2*y^3*s2^5 + (-a),
         Affine Curve over Number Field in a with defining polynomial x^2 - 2
          with a = 1.414213562373095? defined by s0 - 1, 2*z^3 + (-a)*s1^2)
        sage: B[1][0][2]
        Scheme morphism:
          From: Affine Curve over Number Field in a
                with defining polynomial x^2 - 2 with a = 1.414213562373095?
                defined by s2 - 1, 2*x^3 + (-a)*s1^2
          To:   Affine Curve over Number Field in a
                with defining polynomial x^2 - 2 with a = 1.414213562373095?
                defined by s0 - 1, 2*z^3 + (-a)*s1^2
          Defn: Defined on coordinates by sending (x, s1, s2) to
                (x*s2, 1/s2, s1/s2)
        sage: B[1][2][0]
        Scheme morphism:
          From: Affine Curve over Number Field in a
                with defining polynomial x^2 - 2 with a = 1.414213562373095?
                defined by s0 - 1, 2*z^3 + (-a)*s1^2
          To:   Affine Curve over Number Field in a
                with defining polynomial x^2 - 2 with a = 1.414213562373095?
                defined by s2 - 1, 2*x^3 + (-a)*s1^2
          Defn: Defined on coordinates by sending (z, s0, s1) to
                (z*s0, s1/s0, 1/s0)
        sage: B[2]
        (Scheme morphism:
           From: Affine Curve over Number Field in a
                 with defining polynomial x^2 - 2 with a = 1.414213562373095?
                 defined by s2 - 1, 2*x^3 + (-a)*s1^2
           To:   Affine Curve over Number Field in a
                 with defining polynomial x^2 - 2 with a = 1.414213562373095?
                 defined by (-a)*x^5 + y^2, x - z
           Defn: Defined on coordinates by sending (x, s1, s2) to
                 (x, x*s1, x*s2),
         Scheme morphism:
           From: Affine Curve over Number Field in a
                 with defining polynomial x^2 - 2 with a = 1.414213562373095?
                 defined by s0 - s2, 2*y^3*s2^5 + (-a)
           To:   Affine Curve over Number Field in a
                 with defining polynomial x^2 - 2 with a = 1.414213562373095?
                 defined by (-a)*x^5 + y^2, x - z
           Defn: Defined on coordinates by sending (y, s0, s2) to
                 (y*s0, y, y*s2),
         Scheme morphism:
           From: Affine Curve over Number Field in a
                 with defining polynomial x^2 - 2 with a = 1.414213562373095?
                 defined by s0 - 1, 2*z^3 + (-a)*s1^2
           To:   Affine Curve over Number Field in a
                 with defining polynomial x^2 - 2 with a = 1.414213562373095?
                 defined by (-a)*x^5 + y^2, x - z
           Defn: Defined on coordinates by sending (z, s0, s1) to
                 (z*s0, z*s1, z))

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> K = QuadraticField(Integer(2), names=('a',)); (a,) = K._first_ngens(1)
        >>> A = AffineSpace(K, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
        >>> C = Curve([y**Integer(2) - a*x**Integer(5), x - z], A)
        >>> B = C.blowup()
        >>> B[Integer(0)]
        (Affine Curve over Number Field in a with defining polynomial x^2 - 2
          with a = 1.414213562373095? defined by s2 - 1, 2*x^3 + (-a)*s1^2,
         Affine Curve over Number Field in a with defining polynomial x^2 - 2
          with a = 1.414213562373095? defined by s0 - s2, 2*y^3*s2^5 + (-a),
         Affine Curve over Number Field in a with defining polynomial x^2 - 2
          with a = 1.414213562373095? defined by s0 - 1, 2*z^3 + (-a)*s1^2)
        >>> B[Integer(1)][Integer(0)][Integer(2)]
        Scheme morphism:
          From: Affine Curve over Number Field in a
                with defining polynomial x^2 - 2 with a = 1.414213562373095?
                defined by s2 - 1, 2*x^3 + (-a)*s1^2
          To:   Affine Curve over Number Field in a
                with defining polynomial x^2 - 2 with a = 1.414213562373095?
                defined by s0 - 1, 2*z^3 + (-a)*s1^2
          Defn: Defined on coordinates by sending (x, s1, s2) to
                (x*s2, 1/s2, s1/s2)
        >>> B[Integer(1)][Integer(2)][Integer(0)]
        Scheme morphism:
          From: Affine Curve over Number Field in a
                with defining polynomial x^2 - 2 with a = 1.414213562373095?
                defined by s0 - 1, 2*z^3 + (-a)*s1^2
          To:   Affine Curve over Number Field in a
                with defining polynomial x^2 - 2 with a = 1.414213562373095?
                defined by s2 - 1, 2*x^3 + (-a)*s1^2
          Defn: Defined on coordinates by sending (z, s0, s1) to
                (z*s0, s1/s0, 1/s0)
        >>> B[Integer(2)]
        (Scheme morphism:
           From: Affine Curve over Number Field in a
                 with defining polynomial x^2 - 2 with a = 1.414213562373095?
                 defined by s2 - 1, 2*x^3 + (-a)*s1^2
           To:   Affine Curve over Number Field in a
                 with defining polynomial x^2 - 2 with a = 1.414213562373095?
                 defined by (-a)*x^5 + y^2, x - z
           Defn: Defined on coordinates by sending (x, s1, s2) to
                 (x, x*s1, x*s2),
         Scheme morphism:
           From: Affine Curve over Number Field in a
                 with defining polynomial x^2 - 2 with a = 1.414213562373095?
                 defined by s0 - s2, 2*y^3*s2^5 + (-a)
           To:   Affine Curve over Number Field in a
                 with defining polynomial x^2 - 2 with a = 1.414213562373095?
                 defined by (-a)*x^5 + y^2, x - z
           Defn: Defined on coordinates by sending (y, s0, s2) to
                 (y*s0, y, y*s2),
         Scheme morphism:
           From: Affine Curve over Number Field in a
                 with defining polynomial x^2 - 2 with a = 1.414213562373095?
                 defined by s0 - 1, 2*z^3 + (-a)*s1^2
           To:   Affine Curve over Number Field in a
                 with defining polynomial x^2 - 2 with a = 1.414213562373095?
                 defined by (-a)*x^5 + y^2, x - z
           Defn: Defined on coordinates by sending (z, s0, s1) to
                 (z*s0, z*s1, z))

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = A.curve((y - 3/2)^3 - (x + 2)^5 - (x + 2)^6)
        sage: Q = A([-2,3/2])
        sage: C.blowup(Q)
        ((Affine Plane Curve over Rational Field
           defined by x^3 - s1^3 + 7*x^2 + 16*x + 12,
          Affine Plane Curve over Rational Field
           defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5
                      + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8),
         ([Scheme endomorphism of Affine Plane Curve over Rational Field
            defined by x^3 - s1^3 + 7*x^2 + 16*x + 12
             Defn: Defined on coordinates by sending (x, s1) to (x, s1),
           Scheme morphism:
             From: Affine Plane Curve over Rational Field
                   defined by x^3 - s1^3 + 7*x^2 + 16*x + 12
             To:   Affine Plane Curve over Rational Field
                   defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5
                              + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8
             Defn: Defined on coordinates by sending (x, s1) to
                   (x*s1 + 2*s1 + 3/2, 1/s1)],
          [Scheme morphism:
             From: Affine Plane Curve over Rational Field
                   defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5
                              + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8
             To:   Affine Plane Curve over Rational Field
                   defined by x^3 - s1^3 + 7*x^2 + 16*x + 12
             Defn: Defined on coordinates by sending (y, s0) to
                   (y*s0 - 3/2*s0 - 2, 1/s0),
           Scheme endomorphism of Affine Plane Curve over Rational Field
            defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5 + 54*y*s0^6
                       - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8
             Defn: Defined on coordinates by sending (y, s0) to (y, s0)]),
         (Scheme morphism:
            From: Affine Plane Curve over Rational Field
                  defined by x^3 - s1^3 + 7*x^2 + 16*x + 12
            To:   Affine Plane Curve over Rational Field
                  defined by -x^6 - 13*x^5 - 70*x^4 - 200*x^3 + y^3
                             - 320*x^2 - 9/2*y^2 - 272*x + 27/4*y - 795/8
            Defn: Defined on coordinates by sending (x, s1) to
                  (x, x*s1 + 2*s1 + 3/2),
          Scheme morphism:
            From: Affine Plane Curve over Rational Field
                  defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5
                             + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8
            To:   Affine Plane Curve over Rational Field
                  defined by -x^6 - 13*x^5 - 70*x^4 - 200*x^3 + y^3
                             - 320*x^2 - 9/2*y^2 - 272*x + 27/4*y - 795/8
            Defn: Defined on coordinates by sending (y, s0) to
                  (y*s0 - 3/2*s0 - 2, y)))

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = A.curve((y - Integer(3)/Integer(2))**Integer(3) - (x + Integer(2))**Integer(5) - (x + Integer(2))**Integer(6))
        >>> Q = A([-Integer(2),Integer(3)/Integer(2)])
        >>> C.blowup(Q)
        ((Affine Plane Curve over Rational Field
           defined by x^3 - s1^3 + 7*x^2 + 16*x + 12,
          Affine Plane Curve over Rational Field
           defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5
                      + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8),
         ([Scheme endomorphism of Affine Plane Curve over Rational Field
            defined by x^3 - s1^3 + 7*x^2 + 16*x + 12
             Defn: Defined on coordinates by sending (x, s1) to (x, s1),
           Scheme morphism:
             From: Affine Plane Curve over Rational Field
                   defined by x^3 - s1^3 + 7*x^2 + 16*x + 12
             To:   Affine Plane Curve over Rational Field
                   defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5
                              + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8
             Defn: Defined on coordinates by sending (x, s1) to
                   (x*s1 + 2*s1 + 3/2, 1/s1)],
          [Scheme morphism:
             From: Affine Plane Curve over Rational Field
                   defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5
                              + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8
             To:   Affine Plane Curve over Rational Field
                   defined by x^3 - s1^3 + 7*x^2 + 16*x + 12
             Defn: Defined on coordinates by sending (y, s0) to
                   (y*s0 - 3/2*s0 - 2, 1/s0),
           Scheme endomorphism of Affine Plane Curve over Rational Field
            defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5 + 54*y*s0^6
                       - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8
             Defn: Defined on coordinates by sending (y, s0) to (y, s0)]),
         (Scheme morphism:
            From: Affine Plane Curve over Rational Field
                  defined by x^3 - s1^3 + 7*x^2 + 16*x + 12
            To:   Affine Plane Curve over Rational Field
                  defined by -x^6 - 13*x^5 - 70*x^4 - 200*x^3 + y^3
                             - 320*x^2 - 9/2*y^2 - 272*x + 27/4*y - 795/8
            Defn: Defined on coordinates by sending (x, s1) to
                  (x, x*s1 + 2*s1 + 3/2),
          Scheme morphism:
            From: Affine Plane Curve over Rational Field
                  defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5
                             + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8
            To:   Affine Plane Curve over Rational Field
                  defined by -x^6 - 13*x^5 - 70*x^4 - 200*x^3 + y^3
                             - 320*x^2 - 9/2*y^2 - 272*x + 27/4*y - 795/8
            Defn: Defined on coordinates by sending (y, s0) to
                  (y*s0 - 3/2*s0 - 2, y)))

    Sage

        sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
        sage: C = A.curve([((x + 1)^2 + y^2)^3 - 4*(x + 1)^2*y^2, y - z, w - 4])
        sage: Q = C([-1,0,0,4])
        sage: B = C.blowup(Q)
        sage: B[0]
        (Affine Curve over Rational Field defined by s3, s1 - s2,
          x^2*s2^6 + 2*x*s2^6 + 3*x^2*s2^4 + s2^6 + 6*x*s2^4
          + 3*x^2*s2^2 + 3*s2^4 + 6*x*s2^2 + x^2 - s2^2 + 2*x + 1,
         Affine Curve over Rational Field defined by s3, s2 - 1,
          y^2*s0^6 + 3*y^2*s0^4 + 3*y^2*s0^2 + y^2 - 4*s0^2,
         Affine Curve over Rational Field defined by s3, s1 - 1,
          z^2*s0^6 + 3*z^2*s0^4 + 3*z^2*s0^2 + z^2 - 4*s0^2,
         Closed subscheme of Affine Space of dimension 4 over Rational Field
          defined by: 1)
        sage: Q = A([6,2,3,1])
        sage: B = C.blowup(Q)
        Traceback (most recent call last):
        ...
        TypeError: (=(6, 2, 3, 1)) must be a point on this curve

    [Python]

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(4), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = A._first_ngens(4)
        >>> C = A.curve([((x + Integer(1))**Integer(2) + y**Integer(2))**Integer(3) - Integer(4)*(x + Integer(1))**Integer(2)*y**Integer(2), y - z, w - Integer(4)])
        >>> Q = C([-Integer(1),Integer(0),Integer(0),Integer(4)])
        >>> B = C.blowup(Q)
        >>> B[Integer(0)]
        (Affine Curve over Rational Field defined by s3, s1 - s2,
          x^2*s2^6 + 2*x*s2^6 + 3*x^2*s2^4 + s2^6 + 6*x*s2^4
          + 3*x^2*s2^2 + 3*s2^4 + 6*x*s2^2 + x^2 - s2^2 + 2*x + 1,
         Affine Curve over Rational Field defined by s3, s2 - 1,
          y^2*s0^6 + 3*y^2*s0^4 + 3*y^2*s0^2 + y^2 - 4*s0^2,
         Affine Curve over Rational Field defined by s3, s1 - 1,
          z^2*s0^6 + 3*z^2*s0^4 + 3*z^2*s0^2 + z^2 - 4*s0^2,
         Closed subscheme of Affine Space of dimension 4 over Rational Field
          defined by: 1)
        >>> Q = A([Integer(6),Integer(2),Integer(3),Integer(1)])
        >>> B = C.blowup(Q)
        Traceback (most recent call last):
        ...
        TypeError: (=(6, 2, 3, 1)) must be a point on this curve

    Sage

        sage: # needs sage.rings.number_field
        sage: A.<x,y> = AffineSpace(QuadraticField(-1), 2)
        sage: C = A.curve([y^2 + x^2])
        sage: C.blowup()
        Traceback (most recent call last):
        ...
        TypeError: this curve must be irreducible

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> A = AffineSpace(QuadraticField(-Integer(1)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = A.curve([y**Integer(2) + x**Integer(2)])
        >>> C.blowup()
        Traceback (most recent call last):
        ...
        TypeError: this curve must be irreducible

  plane_projection(AP=None)

    Return a projection of this curve into an affine plane so that the image of the projection is a plane curve.

    INPUT:

    - `AP` – (default: `None`) the affine plane to project this curve into. This space must be defined over the same base field as this curve, and must have dimension two. This space will be constructed if not specified.

    OUTPUT: a tuple of

    - a scheme morphism from this curve into an affine plane

    - the plane curve that defines the image of that morphism

    EXAMPLES:

    Sage

        sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
        sage: C = Curve([x^2 - y*z*w, z^3 - w, w + x*y - 3*z^3], A)
        sage: C.plane_projection()
        (Scheme morphism:
          From: Affine Curve over Rational Field defined by
                -y*z*w + x^2, z^3 - w, -3*z^3 + x*y + w
          To:   Affine Space of dimension 2 over Rational Field
          Defn: Defined on coordinates by sending (x, y, z, w) to (x, y),
         Affine Plane Curve over Rational Field defined by
         x0^2*x1^7 - 16*x0^4)

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(4), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = A._first_ngens(4)
        >>> C = Curve([x**Integer(2) - y*z*w, z**Integer(3) - w, w + x*y - Integer(3)*z**Integer(3)], A)
        >>> C.plane_projection()
        (Scheme morphism:
          From: Affine Curve over Rational Field defined by
                -y*z*w + x^2, z^3 - w, -3*z^3 + x*y + w
          To:   Affine Space of dimension 2 over Rational Field
          Defn: Defined on coordinates by sending (x, y, z, w) to (x, y),
         Affine Plane Curve over Rational Field defined by
         x0^2*x1^7 - 16*x0^4)

    Sage

        sage: # needs sage.rings.number_field
        sage: R.<a> = QQ[]
        sage: K.<b> = NumberField(a^2 + 2)
        sage: A.<x,y,z> = AffineSpace(K, 3)
        sage: C = A.curve([x - b, y - 2])
        sage: B.<a,b> = AffineSpace(K, 2)
        sage: proj1 = C.plane_projection(AP=B)
        sage: proj1
        (Scheme morphism:
           From: Affine Curve over Number Field in b
                 with defining polynomial a^2 + 2 defined by x + (-b), y - 2
           To:   Affine Space of dimension 2 over Number Field in b
                 with defining polynomial a^2 + 2
           Defn: Defined on coordinates by sending (x, y, z) to
                 (x, z),
         Affine Plane Curve over Number Field in b
         with defining polynomial a^2 + 2 defined by a + (-b))
        sage: proj1[1].ambient_space() is B
        True
        sage: proj2 = C.plane_projection()
        sage: proj2[1].ambient_space() is B
        False

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> R = QQ['a']; (a,) = R._first_ngens(1)
        >>> K = NumberField(a**Integer(2) + Integer(2), names=('b',)); (b,) = K._first_ngens(1)
        >>> A = AffineSpace(K, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
        >>> C = A.curve([x - b, y - Integer(2)])
        >>> B = AffineSpace(K, Integer(2), names=('a', 'b',)); (a, b,) = B._first_ngens(2)
        >>> proj1 = C.plane_projection(AP=B)
        >>> proj1
        (Scheme morphism:
           From: Affine Curve over Number Field in b
                 with defining polynomial a^2 + 2 defined by x + (-b), y - 2
           To:   Affine Space of dimension 2 over Number Field in b
                 with defining polynomial a^2 + 2
           Defn: Defined on coordinates by sending (x, y, z) to
                 (x, z),
         Affine Plane Curve over Number Field in b
         with defining polynomial a^2 + 2 defined by a + (-b))
        >>> proj1[Integer(1)].ambient_space() is B
        True
        >>> proj2 = C.plane_projection()
        >>> proj2[Integer(1)].ambient_space() is B
        False

  projection(indices, AS=None)

    Return the projection of this curve onto the coordinates specified by `indices`.

    INPUT:

    - `indices` – list or tuple of distinct integers specifying the indices of the coordinates to use in the projection. Can also be a list or tuple consisting of variables of the coordinate ring of the ambient space of this curve. If integers are used to specify the coordinates, 0 denotes the first coordinate. The length of `indices` must be between two and one less than the dimension of the ambient space of this curve, inclusive.

    - `AS` – (default: `None`) the affine space the projected curve will be defined in. This space must be defined over the same base field as this curve, and must have dimension equal to the length of `indices`. This space is constructed if not specified.

    OUTPUT: a tuple of

    - a scheme morphism from this curve to affine space of dimension equal to the number of coordinates specified in `indices`

    - the affine subscheme that is the image of that morphism. If the image is a curve, the second element of the tuple will be a curve.

    EXAMPLES:

    Sage

        sage: A.<x,y,z> = AffineSpace(QQ, 3)
        sage: C = Curve([y^7 - x^2 + x^3 - 2*z, z^2 - x^7 - y^2], A)
        sage: C.projection([0,1])
        (Scheme morphism:
           From: Affine Curve over Rational Field
                 defined by y^7 + x^3 - x^2 - 2*z, -x^7 - y^2 + z^2
           To:   Affine Space of dimension 2 over Rational Field
           Defn: Defined on coordinates by sending (x, y, z) to
                 (x, y),
         Affine Plane Curve over Rational Field defined by x1^14 + 2*x0^3*x1^7 -
        2*x0^2*x1^7 - 4*x0^7 + x0^6 - 2*x0^5 + x0^4 - 4*x1^2)
        sage: C.projection([0,1,3,4])
        Traceback (most recent call last):
        ...
        ValueError: (=[0, 1, 3, 4]) must be a list or tuple of length between 2
        and (=2), inclusive

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
        >>> C = Curve([y**Integer(7) - x**Integer(2) + x**Integer(3) - Integer(2)*z, z**Integer(2) - x**Integer(7) - y**Integer(2)], A)
        >>> C.projection([Integer(0),Integer(1)])
        (Scheme morphism:
           From: Affine Curve over Rational Field
                 defined by y^7 + x^3 - x^2 - 2*z, -x^7 - y^2 + z^2
           To:   Affine Space of dimension 2 over Rational Field
           Defn: Defined on coordinates by sending (x, y, z) to
                 (x, y),
         Affine Plane Curve over Rational Field defined by x1^14 + 2*x0^3*x1^7 -
        2*x0^2*x1^7 - 4*x0^7 + x0^6 - 2*x0^5 + x0^4 - 4*x1^2)
        >>> C.projection([Integer(0),Integer(1),Integer(3),Integer(4)])
        Traceback (most recent call last):
        ...
        ValueError: (=[0, 1, 3, 4]) must be a list or tuple of length between 2
        and (=2), inclusive

    Sage

        sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
        sage: C = Curve([x - 2, y - 3, z - 1], A)
        sage: B.<a,b,c> = AffineSpace(QQ, 3)
        sage: C.projection([0,1,2], AS=B)
        (Scheme morphism:
           From: Affine Curve over Rational Field defined by x - 2, y - 3, z - 1
           To:   Affine Space of dimension 3 over Rational Field
           Defn: Defined on coordinates by sending (x, y, z, w) to
                 (x, y, z),
         Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
           c - 1,
           b - 3,
           a - 2)

    [Python]

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(4), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = A._first_ngens(4)
        >>> C = Curve([x - Integer(2), y - Integer(3), z - Integer(1)], A)
        >>> B = AffineSpace(QQ, Integer(3), names=('a', 'b', 'c',)); (a, b, c,) = B._first_ngens(3)
        >>> C.projection([Integer(0),Integer(1),Integer(2)], AS=B)
        (Scheme morphism:
           From: Affine Curve over Rational Field defined by x - 2, y - 3, z - 1
           To:   Affine Space of dimension 3 over Rational Field
           Defn: Defined on coordinates by sending (x, y, z, w) to
                 (x, y, z),
         Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
           c - 1,
           b - 3,
           a - 2)

    Sage

        sage: A.<x,y,z,w,u> = AffineSpace(GF(11), 5)
        sage: C = Curve([x^3 - 5*y*z + u^2, x - y^2 + 3*z^2,
        ....:            w^2 + 2*u^3*y, y - u^2 + z*x], A)
        sage: B.<a,b,c> = AffineSpace(GF(11), 3)
        sage: proj1 = C.projection([1,2,4], AS=B); proj1
        (Scheme morphism:
           From: Affine Curve over Finite Field of size 11 defined by x^3 -
                 5*y*z + u^2, -y^2 + 3*z^2 + x, 2*y*u^3 + w^2, x*z - u^2 + y
           To:   Affine Space of dimension 3 over Finite Field of size 11
           Defn: Defined on coordinates by sending (x, y, z, w, u) to
                 (y, z, u),
         Affine Curve over Finite Field of size 11 defined by a^2*b - 3*b^3 -
        c^2 + a, c^6 - 5*a*b^4 + b^3*c^2 - 3*a*c^4 + 3*a^2*c^2 - a^3, a^2*c^4 -
        3*b^2*c^4 - 2*a^3*c^2 - 5*a*b^2*c^2 + a^4 - 5*a*b^3 + 2*b^4 + b^2*c^2 -
        3*b*c^2 + 3*a*b, a^4*c^2 + 2*b^4*c^2 - a^5 - 2*a*b^4 + 5*b*c^4 + a*b*c^2
        - 5*a*b^2 + 4*b^3 + b*c^2 + 5*c^2 - 5*a, a^6 - 5*b^6 - 5*b^3*c^2 +
        5*a*b^3 + 2*c^4 - 4*a*c^2 + 2*a^2 - 5*a*b + c^2)
        sage: proj1[1].ambient_space() is B
        True
        sage: proj2 = C.projection([1,2,4])
        sage: proj2[1].ambient_space() is B
        False
        sage: C.projection([1,2,3,5], AS=B)
        Traceback (most recent call last):
        ...
        TypeError: (=Affine Space of dimension 3 over Finite Field of size 11)
        must have dimension (=4)

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(GF(Integer(11)), Integer(5), names=('x', 'y', 'z', 'w', 'u',)); (x, y, z, w, u,) = A._first_ngens(5)
        >>> C = Curve([x**Integer(3) - Integer(5)*y*z + u**Integer(2), x - y**Integer(2) + Integer(3)*z**Integer(2),
        ...            w**Integer(2) + Integer(2)*u**Integer(3)*y, y - u**Integer(2) + z*x], A)
        >>> B = AffineSpace(GF(Integer(11)), Integer(3), names=('a', 'b', 'c',)); (a, b, c,) = B._first_ngens(3)
        >>> proj1 = C.projection([Integer(1),Integer(2),Integer(4)], AS=B); proj1
        (Scheme morphism:
           From: Affine Curve over Finite Field of size 11 defined by x^3 -
                 5*y*z + u^2, -y^2 + 3*z^2 + x, 2*y*u^3 + w^2, x*z - u^2 + y
           To:   Affine Space of dimension 3 over Finite Field of size 11
           Defn: Defined on coordinates by sending (x, y, z, w, u) to
                 (y, z, u),
         Affine Curve over Finite Field of size 11 defined by a^2*b - 3*b^3 -
        c^2 + a, c^6 - 5*a*b^4 + b^3*c^2 - 3*a*c^4 + 3*a^2*c^2 - a^3, a^2*c^4 -
        3*b^2*c^4 - 2*a^3*c^2 - 5*a*b^2*c^2 + a^4 - 5*a*b^3 + 2*b^4 + b^2*c^2 -
        3*b*c^2 + 3*a*b, a^4*c^2 + 2*b^4*c^2 - a^5 - 2*a*b^4 + 5*b*c^4 + a*b*c^2
        - 5*a*b^2 + 4*b^3 + b*c^2 + 5*c^2 - 5*a, a^6 - 5*b^6 - 5*b^3*c^2 +
        5*a*b^3 + 2*c^4 - 4*a*c^2 + 2*a^2 - 5*a*b + c^2)
        >>> proj1[Integer(1)].ambient_space() is B
        True
        >>> proj2 = C.projection([Integer(1),Integer(2),Integer(4)])
        >>> proj2[Integer(1)].ambient_space() is B
        False
        >>> C.projection([Integer(1),Integer(2),Integer(3),Integer(5)], AS=B)
        Traceback (most recent call last):
        ...
        TypeError: (=Affine Space of dimension 3 over Finite Field of size 11)
        must have dimension (=4)

    Sage

        sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
        sage: C = A.curve([x*y - z^3, x*z - w^3, w^2 - x^3])
        sage: C.projection([y,z])
        (Scheme morphism:
           From: Affine Curve over Rational Field defined by
                 -z^3 + x*y, -w^3 + x*z, -x^3 + w^2
           To:   Affine Space of dimension 2 over Rational Field
           Defn: Defined on coordinates by sending (x, y, z, w) to (y, z),
         Affine Plane Curve over Rational Field defined by x1^23 - x0^7*x1^4)
        sage: B.<x,y,z> = AffineSpace(QQ, 3)
        sage: C.projection([x,y,z], AS=B)
        (Scheme morphism:
           From: Affine Curve over Rational Field defined by
                 -z^3 + x*y, -w^3 + x*z, -x^3 + w^2
           To:   Affine Space of dimension 3 over Rational Field
           Defn: Defined on coordinates by sending (x, y, z, w) to
                 (x, y, z),
         Affine Curve over Rational Field defined by
         z^3 - x*y, x^8 - x*z^2, x^7*z^2 - x*y*z)
        sage: C.projection([y,z,z])
        Traceback (most recent call last):
        ...
        ValueError: (=[y, z, z]) must be a list or tuple of distinct indices or
        variables

    [Python]

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(4), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = A._first_ngens(4)
        >>> C = A.curve([x*y - z**Integer(3), x*z - w**Integer(3), w**Integer(2) - x**Integer(3)])
        >>> C.projection([y,z])
        (Scheme morphism:
           From: Affine Curve over Rational Field defined by
                 -z^3 + x*y, -w^3 + x*z, -x^3 + w^2
           To:   Affine Space of dimension 2 over Rational Field
           Defn: Defined on coordinates by sending (x, y, z, w) to (y, z),
         Affine Plane Curve over Rational Field defined by x1^23 - x0^7*x1^4)
        >>> B = AffineSpace(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = B._first_ngens(3)
        >>> C.projection([x,y,z], AS=B)
        (Scheme morphism:
           From: Affine Curve over Rational Field defined by
                 -z^3 + x*y, -w^3 + x*z, -x^3 + w^2
           To:   Affine Space of dimension 3 over Rational Field
           Defn: Defined on coordinates by sending (x, y, z, w) to
                 (x, y, z),
         Affine Curve over Rational Field defined by
         z^3 - x*y, x^8 - x*z^2, x^7*z^2 - x*y*z)
        >>> C.projection([y,z,z])
        Traceback (most recent call last):
        ...
        ValueError: (=[y, z, z]) must be a list or tuple of distinct indices or
        variables

  resolution_of_singularities(extend=False)

    Return a nonsingular model for this affine curve created by blowing up its singular points.

    The nonsingular model is given as a collection of affine patches that cover it. If `extend` is `False` and if the base field is a number field, or if the base field is a finite field, the model returned may have singularities with coordinates not contained in the base field. An error is returned if this curve is already nonsingular, or if it has no singular points over its base field. This curve must be irreducible, and must be defined over a number field or finite field.

    INPUT:

    - `extend` – boolean (default: `False`); specifies whether to extend the base field when necessary to find all singular points when this curve is defined over a number field. If `extend` is `False`, then only singularities with coordinates in the base field of this curve will be resolved. However, setting `extend` to `True` will slow down computations.

    OUTPUT: a tuple of

    - a tuple of curves in affine space of the same dimension as the ambient space of this curve, which represent affine patches of the resolution of singularities.

    - a tuple of tuples such that the j-th element of the i-th tuple is the transition map from the i-th patch to the j-th patch.

    - a tuple consisting of birational maps from the patches back to the original curve that were created by composing the projection maps generated from the blow up computations. There the i-th element will be a map from the i-th patch.

    EXAMPLES:

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve([y^2 - x^3], A)
        sage: C.resolution_of_singularities()
        ((Affine Plane Curve over Rational Field defined by s1^2 - x,
          Affine Plane Curve over Rational Field defined by y*s0^3 - 1),
         ((Scheme endomorphism of Affine Plane Curve over Rational Field
            defined by s1^2 - x
             Defn: Defined on coordinates by sending (x, s1) to (x, s1),
           Scheme morphism:
             From: Affine Plane Curve over Rational Field defined by s1^2 - x
             To:   Affine Plane Curve over Rational Field defined by y*s0^3 - 1
             Defn: Defined on coordinates by sending (x, s1) to (x*s1, 1/s1)),
          (Scheme morphism:
             From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1
             To:   Affine Plane Curve over Rational Field defined by s1^2 - x
             Defn: Defined on coordinates by sending (y, s0) to (y*s0, 1/s0),
           Scheme endomorphism of Affine Plane Curve over Rational Field
            defined by y*s0^3 - 1
             Defn: Defined on coordinates by sending (y, s0) to (y, s0))),
         (Scheme morphism:
            From: Affine Plane Curve over Rational Field defined by s1^2 - x
            To:   Affine Plane Curve over Rational Field defined by -x^3 + y^2
            Defn: Defined on coordinates by sending (x, s1) to (x, x*s1),
          Scheme morphism:
            From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1
            To:   Affine Plane Curve over Rational Field defined by -x^3 + y^2
            Defn: Defined on coordinates by sending (y, s0) to (y*s0, y)))

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([y**Integer(2) - x**Integer(3)], A)
        >>> C.resolution_of_singularities()
        ((Affine Plane Curve over Rational Field defined by s1^2 - x,
          Affine Plane Curve over Rational Field defined by y*s0^3 - 1),
         ((Scheme endomorphism of Affine Plane Curve over Rational Field
            defined by s1^2 - x
             Defn: Defined on coordinates by sending (x, s1) to (x, s1),
           Scheme morphism:
             From: Affine Plane Curve over Rational Field defined by s1^2 - x
             To:   Affine Plane Curve over Rational Field defined by y*s0^3 - 1
             Defn: Defined on coordinates by sending (x, s1) to (x*s1, 1/s1)),
          (Scheme morphism:
             From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1
             To:   Affine Plane Curve over Rational Field defined by s1^2 - x
             Defn: Defined on coordinates by sending (y, s0) to (y*s0, 1/s0),
           Scheme endomorphism of Affine Plane Curve over Rational Field
            defined by y*s0^3 - 1
             Defn: Defined on coordinates by sending (y, s0) to (y, s0))),
         (Scheme morphism:
            From: Affine Plane Curve over Rational Field defined by s1^2 - x
            To:   Affine Plane Curve over Rational Field defined by -x^3 + y^2
            Defn: Defined on coordinates by sending (x, s1) to (x, x*s1),
          Scheme morphism:
            From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1
            To:   Affine Plane Curve over Rational Field defined by -x^3 + y^2
            Defn: Defined on coordinates by sending (y, s0) to (y*s0, y)))

    Sage

        sage: # needs sage.rings.number_field
        sage: set_verbose(-1)
        sage: K.<a> = QuadraticField(3)
        sage: A.<x,y> = AffineSpace(K, 2)
        sage: C = A.curve(x^4 + 2*x^2 + a*y^3 + 1)
        sage: C.resolution_of_singularities(extend=True)[0]         # long time (2 s)
        (Affine Plane Curve over Number Field in a0
          with defining polynomial y^4 - 4*y^2 + 16
          defined by 24*x^2*ss1^3 + 24*ss1^3 + (a0^3 - 8*a0),
         Affine Plane Curve over Number Field in a0
          with defining polynomial y^4 - 4*y^2 + 16
          defined by 24*s1^2*ss0 + (a0^3 - 8*a0)*ss0^2 + (-6*a0^3)*s1,
         Affine Plane Curve over Number Field in a0
          with defining polynomial y^4 - 4*y^2 + 16
          defined by 8*y^2*s0^4 + (4*a0^3)*y*s0^3 - 32*s0^2 + (a0^3 - 8*a0)*y)

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> set_verbose(-Integer(1))
        >>> K = QuadraticField(Integer(3), names=('a',)); (a,) = K._first_ngens(1)
        >>> A = AffineSpace(K, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = A.curve(x**Integer(4) + Integer(2)*x**Integer(2) + a*y**Integer(3) + Integer(1))
        >>> C.resolution_of_singularities(extend=True)[Integer(0)]         # long time (2 s)
        (Affine Plane Curve over Number Field in a0
          with defining polynomial y^4 - 4*y^2 + 16
          defined by 24*x^2*ss1^3 + 24*ss1^3 + (a0^3 - 8*a0),
         Affine Plane Curve over Number Field in a0
          with defining polynomial y^4 - 4*y^2 + 16
          defined by 24*s1^2*ss0 + (a0^3 - 8*a0)*ss0^2 + (-6*a0^3)*s1,
         Affine Plane Curve over Number Field in a0
          with defining polynomial y^4 - 4*y^2 + 16
          defined by 8*y^2*s0^4 + (4*a0^3)*y*s0^3 - 32*s0^2 + (a0^3 - 8*a0)*y)

    Sage

        sage: A.<x,y,z> = AffineSpace(GF(5), 3)
        sage: C = Curve([y - x^3, (z - 2)^2 - y^3 - x^3], A)
        sage: R = C.resolution_of_singularities()
        sage: R[0]
        (Affine Curve over Finite Field of size 5
          defined by x^2 - s1, s1^4 - x*s2^2 + s1, x*s1^3 - s2^2 + x,
         Affine Curve over Finite Field of size 5
          defined by y*s2^2 - y^2 - 1, s2^4 - s0^3 - y^2 - 2, y*s0^3 - s2^2 + y,
         Affine Curve over Finite Field of size 5
          defined by s0^3*s1 + z*s1^3 + s1^4 - 2*s1^3 - 1,
                     z*s0^3 + z*s1^3 - 2*s0^3 - 2*s1^3 - 1,
                     z^2*s1^3 + z*s1^3 - s1^3 - z + s1 + 2)

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(GF(Integer(5)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
        >>> C = Curve([y - x**Integer(3), (z - Integer(2))**Integer(2) - y**Integer(3) - x**Integer(3)], A)
        >>> R = C.resolution_of_singularities()
        >>> R[Integer(0)]
        (Affine Curve over Finite Field of size 5
          defined by x^2 - s1, s1^4 - x*s2^2 + s1, x*s1^3 - s2^2 + x,
         Affine Curve over Finite Field of size 5
          defined by y*s2^2 - y^2 - 1, s2^4 - s0^3 - y^2 - 2, y*s0^3 - s2^2 + y,
         Affine Curve over Finite Field of size 5
          defined by s0^3*s1 + z*s1^3 + s1^4 - 2*s1^3 - 1,
                     z*s0^3 + z*s1^3 - 2*s0^3 - 2*s1^3 - 1,
                     z^2*s1^3 + z*s1^3 - s1^3 - z + s1 + 2)

    Sage

        sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
        sage: C = A.curve([((x - 2)^2 + y^2)^2 - (x - 2)^2 - y^2 + (x - 2)^3,
        ....:              z - y - 7, w - 4])
        sage: B = C.resolution_of_singularities()
        sage: B[0]
        (Affine Curve over Rational Field defined by s3, s1 - s2,
          x^2*s2^4 - 4*x*s2^4 + 2*x^2*s2^2 + 4*s2^4 - 8*x*s2^2
          + x^2 + 7*s2^2 - 3*x + 1,
         Affine Curve over Rational Field defined by s3, s2 - 1,
          y^2*s0^4 + 2*y^2*s0^2 + y*s0^3 + y^2 - s0^2 - 1,
         Affine Curve over Rational Field defined by s3, s1 - 1,
          z^2*s0^4 - 14*z*s0^4 + 2*z^2*s0^2 + z*s0^3 + 49*s0^4
          - 28*z*s0^2 - 7*s0^3 + z^2 + 97*s0^2 - 14*z + 48,
         Closed subscheme of Affine Space of dimension 4 over Rational Field
          defined by: 1)

    [Python]

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(4), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = A._first_ngens(4)
        >>> C = A.curve([((x - Integer(2))**Integer(2) + y**Integer(2))**Integer(2) - (x - Integer(2))**Integer(2) - y**Integer(2) + (x - Integer(2))**Integer(3),
        ...              z - y - Integer(7), w - Integer(4)])
        >>> B = C.resolution_of_singularities()
        >>> B[Integer(0)]
        (Affine Curve over Rational Field defined by s3, s1 - s2,
          x^2*s2^4 - 4*x*s2^4 + 2*x^2*s2^2 + 4*s2^4 - 8*x*s2^2
          + x^2 + 7*s2^2 - 3*x + 1,
         Affine Curve over Rational Field defined by s3, s2 - 1,
          y^2*s0^4 + 2*y^2*s0^2 + y*s0^3 + y^2 - s0^2 - 1,
         Affine Curve over Rational Field defined by s3, s1 - 1,
          z^2*s0^4 - 14*z*s0^4 + 2*z^2*s0^2 + z*s0^3 + 49*s0^4
          - 28*z*s0^2 - 7*s0^3 + z^2 + 97*s0^2 - 14*z + 48,
         Closed subscheme of Affine Space of dimension 4 over Rational Field
          defined by: 1)

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve([y - x^2 + 1], A)
        sage: C.resolution_of_singularities()
        Traceback (most recent call last):
        ...
        TypeError: this curve is already nonsingular

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([y - x**Integer(2) + Integer(1)], A)
        >>> C.resolution_of_singularities()
        Traceback (most recent call last):
        ...
        TypeError: this curve is already nonsingular

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = A.curve([(x^2 + y^2 - y - 2)*(y - x^2 + 2) + y^3])
        sage: C.resolution_of_singularities()
        Traceback (most recent call last):
        ...
        TypeError: this curve has no singular points over its base field. If
        working over a number field use extend=True

    [Python]

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = A.curve([(x**Integer(2) + y**Integer(2) - y - Integer(2))*(y - x**Integer(2) + Integer(2)) + y**Integer(3)])
        >>> C.resolution_of_singularities()
        Traceback (most recent call last):
        ...
        TypeError: this curve has no singular points over its base field. If
        working over a number field use extend=True

  tangent_line(p)

    Return the tangent line at the point `p`.

    INPUT:

    - `p` – a rational point of the curve

    EXAMPLES:

    Sage

        sage: A3.<x,y,z> = AffineSpace(3, QQ)
        sage: C = Curve([x + y + z, x^2 - y^2*z^2 + z^3])
        sage: p = C(0,0,0)
        sage: C.tangent_line(p)
        Traceback (most recent call last):
        ...
        ValueError: the curve is not smooth at (0, 0, 0)
        sage: p = C(1,0,-1)
        sage: C.tangent_line(p)
        Affine Curve over Rational Field defined by x + y + z, 2*x + 3*z + 1

    Python

        >>> from sage.all import *
        >>> A3 = AffineSpace(Integer(3), QQ, names=('x', 'y', 'z',)); (x, y, z,) = A3._first_ngens(3)
        >>> C = Curve([x + y + z, x**Integer(2) - y**Integer(2)*z**Integer(2) + z**Integer(3)])
        >>> p = C(Integer(0),Integer(0),Integer(0))
        >>> C.tangent_line(p)
        Traceback (most recent call last):
        ...
        ValueError: the curve is not smooth at (0, 0, 0)
        >>> p = C(Integer(1),Integer(0),-Integer(1))
        >>> C.tangent_line(p)
        Affine Curve over Rational Field defined by x + y + z, 2*x + 3*z + 1

    We check that the tangent line at `p` is the tangent space at `p`, translated to `p`.

    Sage

        sage: Tp = C.tangent_space(p)
        sage: Tp
        Closed subscheme of Affine Space of dimension 3 over Rational Field
         defined by: x + y + z, 2*x + 3*z
        sage: phi = A3.translation(A3.origin(), p)
        sage: T = phi * Tp.embedding_morphism()
        sage: T.image()
        Closed subscheme of Affine Space of dimension 3 over Rational Field
         defined by: -2*y + z + 1, x + y + z
        sage: _ == C.tangent_line(p)
        True

    [Python]

        >>> from sage.all import *
        >>> Tp = C.tangent_space(p)
        >>> Tp
        Closed subscheme of Affine Space of dimension 3 over Rational Field
         defined by: x + y + z, 2*x + 3*z
        >>> phi = A3.translation(A3.origin(), p)
        >>> T = phi * Tp.embedding_morphism()
        >>> T.image()
        Closed subscheme of Affine Space of dimension 3 over Rational Field
         defined by: -2*y + z + 1, x + y + z
        >>> _ == C.tangent_line(p)
        True

class sage.schemes.curves.affine_curve.AffinePlaneCurve(A, f)

  Bases: `AffineCurve`

  Affine plane curves.

  divisor_of_function(r)

    Return the divisor of a function on a curve.

    INPUT:

    - `r` – a rational function on X

    OUTPUT:

    - `list` – the divisor of r represented as a list of coefficients and points. (TODO: This will change to a more structural output in the future.)

    EXAMPLES:

    Sage

        sage: F = GF(5)
        sage: P2 = AffineSpace(2, F, names='xy')
        sage: R = P2.coordinate_ring()
        sage: x, y = R.gens()
        sage: f = y^2 - x^9 - x
        sage: C = Curve(f)
        sage: K = FractionField(R)
        sage: r = 1/x
        sage: C.divisor_of_function(r)      # not implemented (broken)
              [[-1, (0, 0, 1)]]
        sage: r = 1/x^3
        sage: C.divisor_of_function(r)      # not implemented (broken)
              [[-3, (0, 0, 1)]]

    Python

        >>> from sage.all import *
        >>> F = GF(Integer(5))
        >>> P2 = AffineSpace(Integer(2), F, names='xy')
        >>> R = P2.coordinate_ring()
        >>> x, y = R.gens()
        >>> f = y**Integer(2) - x**Integer(9) - x
        >>> C = Curve(f)
        >>> K = FractionField(R)
        >>> r = Integer(1)/x
        >>> C.divisor_of_function(r)      # not implemented (broken)
              [[-1, (0, 0, 1)]]
        >>> r = Integer(1)/x**Integer(3)
        >>> C.divisor_of_function(r)      # not implemented (broken)
              [[-3, (0, 0, 1)]]

  is_ordinary_singularity(P)

    Return whether the singular point `P` of this affine plane curve is an ordinary singularity.

    The point `P` is an ordinary singularity of this curve if it is a singular point, and if the tangents of this curve at `P` are distinct.

    INPUT:

    - `P` – a point on this curve

    OUTPUT:

    `True` or `False` depending on whether `P` is or is not an ordinary singularity of this curve, respectively. An error is raised if `P` is not a singular point of this curve.

    EXAMPLES:

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve([y^2 - x^3], A)
        sage: Q = A([0,0])
        sage: C.is_ordinary_singularity(Q)
        False

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([y**Integer(2) - x**Integer(3)], A)
        >>> Q = A([Integer(0),Integer(0)])
        >>> C.is_ordinary_singularity(Q)
        False

    Sage

        sage: # needs sage.rings.number_field
        sage: R.<a> = QQ[]
        sage: K.<b> = NumberField(a^2 - 3)
        sage: A.<x,y> = AffineSpace(K, 2)
        sage: C = Curve([(x^2 + y^2 - 2*x)^2 - x^2 - y^2], A)
        sage: Q = A([0,0])
        sage: C.is_ordinary_singularity(Q)
        True

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> R = QQ['a']; (a,) = R._first_ngens(1)
        >>> K = NumberField(a**Integer(2) - Integer(3), names=('b',)); (b,) = K._first_ngens(1)
        >>> A = AffineSpace(K, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([(x**Integer(2) + y**Integer(2) - Integer(2)*x)**Integer(2) - x**Integer(2) - y**Integer(2)], A)
        >>> Q = A([Integer(0),Integer(0)])
        >>> C.is_ordinary_singularity(Q)
        True

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = A.curve([x^2*y - y^2*x + y^2 + x^3])
        sage: Q = A([-1,-1])
        sage: C.is_ordinary_singularity(Q)
        Traceback (most recent call last):
        ...
        TypeError: (=(-1, -1)) is not a singular point of (=Affine Plane Curve
        over Rational Field defined by x^3 + x^2*y - x*y^2 + y^2)

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = A.curve([x**Integer(2)*y - y**Integer(2)*x + y**Integer(2) + x**Integer(3)])
        >>> Q = A([-Integer(1),-Integer(1)])
        >>> C.is_ordinary_singularity(Q)
        Traceback (most recent call last):
        ...
        TypeError: (=(-1, -1)) is not a singular point of (=Affine Plane Curve
        over Rational Field defined by x^3 + x^2*y - x*y^2 + y^2)

  is_transverse(C, P)

    Return whether the intersection of this curve with the curve `C` at the point `P` is transverse.

    The intersection at `P` is transverse if `P` is a nonsingular point of both curves, and if the tangents of the curves at `P` are distinct.

    INPUT:

    - `C` – a curve in the ambient space of this curve

    - `P` – a point in the intersection of both curves

    OUTPUT: boolean

    EXAMPLES:

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve([x^2 + y^2 - 1], A)
        sage: D = Curve([x - 1], A)
        sage: Q = A([1,0])
        sage: C.is_transverse(D, Q)
        False

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([x**Integer(2) + y**Integer(2) - Integer(1)], A)
        >>> D = Curve([x - Integer(1)], A)
        >>> Q = A([Integer(1),Integer(0)])
        >>> C.is_transverse(D, Q)
        False

    Sage

        sage: # needs sage.rings.number_field
        sage: R.<a> = QQ[]
        sage: K.<b> = NumberField(a^3 + 2)
        sage: A.<x,y> = AffineSpace(K, 2)
        sage: C = A.curve([x*y])
        sage: D = A.curve([y - b*x])
        sage: Q = A([0,0])
        sage: C.is_transverse(D, Q)
        False

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> R = QQ['a']; (a,) = R._first_ngens(1)
        >>> K = NumberField(a**Integer(3) + Integer(2), names=('b',)); (b,) = K._first_ngens(1)
        >>> A = AffineSpace(K, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = A.curve([x*y])
        >>> D = A.curve([y - b*x])
        >>> Q = A([Integer(0),Integer(0)])
        >>> C.is_transverse(D, Q)
        False

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve([y - x^3], A)
        sage: D = Curve([y + x], A)
        sage: Q = A([0,0])
        sage: C.is_transverse(D, Q)
        True

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([y - x**Integer(3)], A)
        >>> D = Curve([y + x], A)
        >>> Q = A([Integer(0),Integer(0)])
        >>> C.is_transverse(D, Q)
        True

  local_coordinates(pt, n)

    Return local coordinates to precision n at the given point.

    Behaviour is flaky - some choices of \(n\) are worst that others.

    INPUT:

    - `pt` – an F-rational point on X which is not a point of ramification for the projection (x,y) - x

    - `n` – the number of terms desired

    OUTPUT: x = x0 + t y = y0 + power series in t

    EXAMPLES:

    Sage

        sage: F = GF(5)
        sage: pt = (2,3)
        sage: R = PolynomialRing(F, 2, names = ['x','y'])
        sage: x,y = R.gens()
        sage: f = y^2 - x^9 - x
        sage: C = Curve(f)
        sage: C.local_coordinates(pt, 9)
        [t + 2, -2*t^12 - 2*t^11 + 2*t^9 + t^8 - 2*t^7 - 2*t^6 - 2*t^4 + t^3 - 2*t^2 - 2]

    Python

        >>> from sage.all import *
        >>> F = GF(Integer(5))
        >>> pt = (Integer(2),Integer(3))
        >>> R = PolynomialRing(F, Integer(2), names = ['x','y'])
        >>> x,y = R.gens()
        >>> f = y**Integer(2) - x**Integer(9) - x
        >>> C = Curve(f)
        >>> C.local_coordinates(pt, Integer(9))
        [t + 2, -2*t^12 - 2*t^11 + 2*t^9 + t^8 - 2*t^7 - 2*t^6 - 2*t^4 + t^3 - 2*t^2 - 2]

  multiplicity(P)

    Return the multiplicity of this affine plane curve at the point `P`.

    In the special case of affine plane curves, the multiplicity of an affine plane curve at the point (0,0) can be computed as the minimum of the degrees of the homogeneous components of its defining polynomial. To compute the multiplicity of a different point, a linear change of coordinates is used.

    This curve must be defined over a field. An error if raised if `P` is not a point on this curve.

    INPUT:

    - `P` – a point in the ambient space of this curve

    OUTPUT: integer

    EXAMPLES:

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve([y^2 - x^3], A)
        sage: Q1 = A([1,1])
        sage: C.multiplicity(Q1)
        1
        sage: Q2 = A([0,0])
        sage: C.multiplicity(Q2)
        2

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([y**Integer(2) - x**Integer(3)], A)
        >>> Q1 = A([Integer(1),Integer(1)])
        >>> C.multiplicity(Q1)
        1
        >>> Q2 = A([Integer(0),Integer(0)])
        >>> C.multiplicity(Q2)
        2

    Sage

        sage: # needs sage.rings.number_field
        sage: A.<x,y> = AffineSpace(QQbar,2)
        sage: C = Curve([-x^7 + (-7)*x^6 + y^6 + (-21)*x^5 + 12*y^5
        ....:            + (-35)*x^4 + 60*y^4 + (-35)*x^3 + 160*y^3
        ....:            + (-21)*x^2 + 240*y^2 + (-7)*x + 192*y + 63], A)
        sage: Q = A([-1,-2])
        sage: C.multiplicity(Q)
        6

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> A = AffineSpace(QQbar,Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([-x**Integer(7) + (-Integer(7))*x**Integer(6) + y**Integer(6) + (-Integer(21))*x**Integer(5) + Integer(12)*y**Integer(5)
        ...            + (-Integer(35))*x**Integer(4) + Integer(60)*y**Integer(4) + (-Integer(35))*x**Integer(3) + Integer(160)*y**Integer(3)
        ...            + (-Integer(21))*x**Integer(2) + Integer(240)*y**Integer(2) + (-Integer(7))*x + Integer(192)*y + Integer(63)], A)
        >>> Q = A([-Integer(1),-Integer(2)])
        >>> C.multiplicity(Q)
        6

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = A.curve([y^3 - x^3 + x^6])
        sage: Q = A([1,1])
        sage: C.multiplicity(Q)
        Traceback (most recent call last):
        ...
        TypeError: (=(1, 1)) is not a point on (=Affine Plane Curve over
        Rational Field defined by x^6 - x^3 + y^3)

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = A.curve([y**Integer(3) - x**Integer(3) + x**Integer(6)])
        >>> Q = A([Integer(1),Integer(1)])
        >>> C.multiplicity(Q)
        Traceback (most recent call last):
        ...
        TypeError: (=(1, 1)) is not a point on (=Affine Plane Curve over
        Rational Field defined by x^6 - x^3 + y^3)

  plot(*args, **kwds)

    Plot the real points on this affine plane curve.

    INPUT:

    - `*args` – (optional) tuples (variable, minimum, maximum) for plotting dimensions

    - `**kwds` – optional keyword arguments passed on to `implicit_plot`

    EXAMPLES:

    A cuspidal curve:

    Sage

        sage: R.<x, y> = QQ[]
        sage: C = Curve(x^3 - y^2)
        sage: C.plot()                                      # needs sage.plot
        Graphics object consisting of 1 graphics primitive

    Python

        >>> from sage.all import *
        >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
        >>> C = Curve(x**Integer(3) - y**Integer(2))
        >>> C.plot()                                      # needs sage.plot
        Graphics object consisting of 1 graphics primitive

    A 5-nodal curve of degree 11. This example also illustrates some of the optional arguments:

    Sage

        sage: # needs sage.plot
        sage: R.<x, y> = ZZ[]
        sage: C = Curve(32*x^2 - 2097152*y^11 + 1441792*y^9
        ....:            - 360448*y^7 + 39424*y^5 - 1760*y^3 + 22*y - 1)
        sage: C.plot((x, -1, 1), (y, -1, 1), plot_points=400)
        Graphics object consisting of 1 graphics primitive

    [Python]

        >>> from sage.all import *
        >>> # needs sage.plot
        >>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2)
        >>> C = Curve(Integer(32)*x**Integer(2) - Integer(2097152)*y**Integer(11) + Integer(1441792)*y**Integer(9)
        ...            - Integer(360448)*y**Integer(7) + Integer(39424)*y**Integer(5) - Integer(1760)*y**Integer(3) + Integer(22)*y - Integer(1))
        >>> C.plot((x, -Integer(1), Integer(1)), (y, -Integer(1), Integer(1)), plot_points=Integer(400))
        Graphics object consisting of 1 graphics primitive

    A line over \(\mathbf{RR}\):

    Sage

        sage: # needs sage.symbolic sage.plot
        sage: R.<x, y> = RR[]
        sage: C = Curve(R(y - sqrt(2)*x))
        sage: C.plot()
        Graphics object consisting of 1 graphics primitive

    Python

        >>> from sage.all import *
        >>> # needs sage.symbolic sage.plot
        >>> R = RR['x, y']; (x, y,) = R._first_ngens(2)
        >>> C = Curve(R(y - sqrt(Integer(2))*x))
        >>> C.plot()
        Graphics object consisting of 1 graphics primitive

  rational_parameterization()

    Return a rational parameterization of this curve.

    This curve must have rational coefficients and be absolutely irreducible (i.e. irreducible over the algebraic closure of the rational field). The curve must also be rational (have geometric genus zero).

    The rational parameterization may have coefficients in a quadratic extension of the rational field.

    OUTPUT: a birational map between \(\mathbb{A}^{1}\) and this curve, given as a scheme morphism

    EXAMPLES:

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve([y^2 - x], A)
        sage: C.rational_parameterization()
        Scheme morphism:
          From: Affine Space of dimension 1 over Rational Field
          To:   Affine Plane Curve over Rational Field defined by y^2 - x
          Defn: Defined on coordinates by sending (t) to
                (t^2, t)

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([y**Integer(2) - x], A)
        >>> C.rational_parameterization()
        Scheme morphism:
          From: Affine Space of dimension 1 over Rational Field
          To:   Affine Plane Curve over Rational Field defined by y^2 - x
          Defn: Defined on coordinates by sending (t) to
                (t^2, t)

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve([(x^2 + y^2 - 2*x)^2 - x^2 - y^2], A)
        sage: C.rational_parameterization()
        Scheme morphism:
          From: Affine Space of dimension 1 over Rational Field
          To:   Affine Plane Curve over Rational Field defined by x^4 +
        2*x^2*y^2 + y^4 - 4*x^3 - 4*x*y^2 + 3*x^2 - y^2
          Defn: Defined on coordinates by sending (t) to
                ((-12*t^4 + 6*t^3 + 4*t^2 - 2*t)/(-25*t^4 + 40*t^3 - 26*t^2 +
        8*t - 1), (-9*t^4 + 12*t^3 - 4*t + 1)/(-25*t^4 + 40*t^3 - 26*t^2 + 8*t - 1))

    [Python]

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([(x**Integer(2) + y**Integer(2) - Integer(2)*x)**Integer(2) - x**Integer(2) - y**Integer(2)], A)
        >>> C.rational_parameterization()
        Scheme morphism:
          From: Affine Space of dimension 1 over Rational Field
          To:   Affine Plane Curve over Rational Field defined by x^4 +
        2*x^2*y^2 + y^4 - 4*x^3 - 4*x*y^2 + 3*x^2 - y^2
          Defn: Defined on coordinates by sending (t) to
                ((-12*t^4 + 6*t^3 + 4*t^2 - 2*t)/(-25*t^4 + 40*t^3 - 26*t^2 +
        8*t - 1), (-9*t^4 + 12*t^3 - 4*t + 1)/(-25*t^4 + 40*t^3 - 26*t^2 + 8*t - 1))

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve([x^2 + y^2 + 7], A)
        sage: C.rational_parameterization()
        Scheme morphism:
          From: Affine Space of dimension 1 over Number Field in a with defining polynomial a^2 + 7
          To:   Affine Plane Curve over Number Field in a with defining
        polynomial a^2 + 7 defined by x^2 + y^2 + 7
          Defn: Defined on coordinates by sending (t) to
                ((-7*t^2 + 7)/((-a)*t^2 + (-a)), 14*t/((-a)*t^2 + (-a)))

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([x**Integer(2) + y**Integer(2) + Integer(7)], A)
        >>> C.rational_parameterization()
        Scheme morphism:
          From: Affine Space of dimension 1 over Number Field in a with defining polynomial a^2 + 7
          To:   Affine Plane Curve over Number Field in a with defining
        polynomial a^2 + 7 defined by x^2 + y^2 + 7
          Defn: Defined on coordinates by sending (t) to
                ((-7*t^2 + 7)/((-a)*t^2 + (-a)), 14*t/((-a)*t^2 + (-a)))

  tangents(P, factor=True)

    Return the tangents of this affine plane curve at the point `P`.

    The point `P` must be a point on this curve.

    INPUT:

    - `P` – a point on this curve

    - `factor` – boolean (default: `True`); whether to attempt computing the polynomials of the individual tangent lines over the base field of this curve, or to just return the polynomial corresponding to the union of the tangent lines (which requires fewer computations)

    OUTPUT: list of polynomials in the coordinate ring of the ambient space

    EXAMPLES:

    Sage

        sage: # needs sage.rings.number_field
        sage: set_verbose(-1)
        sage: A.<x,y> = AffineSpace(QQbar, 2)
        sage: C = Curve([x^5*y^3 + 2*x^4*y^4 + x^3*y^5 + 3*x^4*y^3
        ....:            + 6*x^3*y^4 + 3*x^2*y^5 + 3*x^3*y^3
        ....:            + 6*x^2*y^4 + 3*x*y^5 + x^5 + 10*x^4*y
        ....:            + 40*x^3*y^2 + 81*x^2*y^3 + 82*x*y^4 + 33*y^5], A)
        sage: Q = A([0,0])
        sage: C.tangents(Q)
        [x + 3.425299577684700?*y,
         x + (1.949159013086856? + 1.179307909383728?*I)*y,
         x + (1.949159013086856? - 1.179307909383728?*I)*y,
         x + (1.338191198070795? + 0.2560234251008043?*I)*y,
         x + (1.338191198070795? - 0.2560234251008043?*I)*y]
        sage: C.tangents(Q, factor=False)
        [120*x^5 + 1200*x^4*y + 4800*x^3*y^2 + 9720*x^2*y^3 + 9840*x*y^4 + 3960*y^5]

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> set_verbose(-Integer(1))
        >>> A = AffineSpace(QQbar, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([x**Integer(5)*y**Integer(3) + Integer(2)*x**Integer(4)*y**Integer(4) + x**Integer(3)*y**Integer(5) + Integer(3)*x**Integer(4)*y**Integer(3)
        ...            + Integer(6)*x**Integer(3)*y**Integer(4) + Integer(3)*x**Integer(2)*y**Integer(5) + Integer(3)*x**Integer(3)*y**Integer(3)
        ...            + Integer(6)*x**Integer(2)*y**Integer(4) + Integer(3)*x*y**Integer(5) + x**Integer(5) + Integer(10)*x**Integer(4)*y
        ...            + Integer(40)*x**Integer(3)*y**Integer(2) + Integer(81)*x**Integer(2)*y**Integer(3) + Integer(82)*x*y**Integer(4) + Integer(33)*y**Integer(5)], A)
        >>> Q = A([Integer(0),Integer(0)])
        >>> C.tangents(Q)
        [x + 3.425299577684700?*y,
         x + (1.949159013086856? + 1.179307909383728?*I)*y,
         x + (1.949159013086856? - 1.179307909383728?*I)*y,
         x + (1.338191198070795? + 0.2560234251008043?*I)*y,
         x + (1.338191198070795? - 0.2560234251008043?*I)*y]
        >>> C.tangents(Q, factor=False)
        [120*x^5 + 1200*x^4*y + 4800*x^3*y^2 + 9720*x^2*y^3 + 9840*x*y^4 + 3960*y^5]

    Sage

        sage: # needs sage.rings.number_field
        sage: R.<a> = QQ[]
        sage: K.<b> = NumberField(a^2 - 3)
        sage: A.<x,y> = AffineSpace(K, 2)
        sage: C = Curve([(x^2 + y^2 - 2*x)^2 - x^2 - y^2], A)
        sage: Q = A([0,0])
        sage: C.tangents(Q)
        [x + (-1/3*b)*y, x + (1/3*b)*y]

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> R = QQ['a']; (a,) = R._first_ngens(1)
        >>> K = NumberField(a**Integer(2) - Integer(3), names=('b',)); (b,) = K._first_ngens(1)
        >>> A = AffineSpace(K, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve([(x**Integer(2) + y**Integer(2) - Integer(2)*x)**Integer(2) - x**Integer(2) - y**Integer(2)], A)
        >>> Q = A([Integer(0),Integer(0)])
        >>> C.tangents(Q)
        [x + (-1/3*b)*y, x + (1/3*b)*y]

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = A.curve([y^2 - x^3 - x^2])
        sage: Q = A([0,0])
        sage: C.tangents(Q)
        [x - y, x + y]

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = A.curve([y**Integer(2) - x**Integer(3) - x**Integer(2)])
        >>> Q = A([Integer(0),Integer(0)])
        >>> C.tangents(Q)
        [x - y, x + y]

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = A.curve([y*x - x^4 + 2*x^2])
        sage: Q = A([1,1])
        sage: C.tangents(Q)
        Traceback (most recent call last):
        ...
        TypeError: (=(1, 1)) is not a point on (=Affine Plane Curve over
        Rational Field defined by -x^4 + 2*x^2 + x*y)

    [Python]

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = A.curve([y*x - x**Integer(4) + Integer(2)*x**Integer(2)])
        >>> Q = A([Integer(1),Integer(1)])
        >>> C.tangents(Q)
        Traceback (most recent call last):
        ...
        TypeError: (=(1, 1)) is not a point on (=Affine Plane Curve over
        Rational Field defined by -x^4 + 2*x^2 + x*y)

class sage.schemes.curves.affine_curve.AffinePlaneCurve_field(A, f)

  Bases: `AffinePlaneCurve`, `AffineCurve_field`

  Affine plane curves over fields.

  braid_monodromy()

    Compute the braid monodromy of a projection of the curve.

    OUTPUT:

    A list of braids. The braids correspond to paths based in the same point; each of this paths is the conjugated of a loop around one of the points in the discriminant of the projection of `self`.

    Note

    The projection over the \(x\) axis is used if there are no vertical asymptotes. Otherwise, a linear change of variables is done to fall into the previous case.

    Note

    This functionality requires the `sirocco` package to be installed.

    EXAMPLES:

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = A.curve((x^2-y^3)*(x+3*y-5))
        sage: C.braid_monodromy()                                   # needs sirocco
        [s1*s0*(s1*s2)^2*s0*s2^2*s0^-1*(s2^-1*s1^-1)^2*s0^-1*s1^-1,
         s1*s0*(s1*s2)^2*(s0*s2^-1*s1*s2*s1*s2^-1)^2*(s2^-1*s1^-1)^2*s0^-1*s1^-1,
         s1*s0*(s1*s2)^2*s2*s1^-1*s2^-1*s1^-1*s0^-1*s1^-1,
         s1*s0*s2*s0^-1*s2*s1^-1]
        sage: T.<t> = QQ[]
        sage: K.<a> = NumberField(t^3 + 2, 'a')
        sage: A.<x, y> = AffineSpace(K, 2)
        sage: Curve(y^2 + a * x).braid_monodromy()
        Traceback (most recent call last):
        ...
        NotImplementedError: the base field must have an embedding to the algebraic field

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = A.curve((x**Integer(2)-y**Integer(3))*(x+Integer(3)*y-Integer(5)))
        >>> C.braid_monodromy()                                   # needs sirocco
        [s1*s0*(s1*s2)^2*s0*s2^2*s0^-1*(s2^-1*s1^-1)^2*s0^-1*s1^-1,
         s1*s0*(s1*s2)^2*(s0*s2^-1*s1*s2*s1*s2^-1)^2*(s2^-1*s1^-1)^2*s0^-1*s1^-1,
         s1*s0*(s1*s2)^2*s2*s1^-1*s2^-1*s1^-1*s0^-1*s1^-1,
         s1*s0*s2*s0^-1*s2*s1^-1]
        >>> T = QQ['t']; (t,) = T._first_ngens(1)
        >>> K = NumberField(t**Integer(3) + Integer(2), 'a', names=('a',)); (a,) = K._first_ngens(1)
        >>> A = AffineSpace(K, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> Curve(y**Integer(2) + a * x).braid_monodromy()
        Traceback (most recent call last):
        ...
        NotImplementedError: the base field must have an embedding to the algebraic field

  fundamental_group(simplified=True, puiseux=True)

    Return a presentation of the fundamental group of the complement of `self`.

    INPUT:

    - `simplified` – boolean (default: `True`); to simplify the presentation

    - `puiseux` – boolean (default: `True`); to decide if the presentation is constructed in the classical way or using Puiseux shortcut

    OUTPUT:

    A presentation with generators \(x_1, \dots, x_d\) and relations. If `puiseux` is `False` the relations are \((x_j\cdot \tau)\cdot x_j^{-1}\) for \(1\leq j<d\) and \(tau\) a braid in the braid monodromy; finally the presentation is simplified. If `puiseux` is `True`, each \(tau\) is decomposed as \(\alpha^{-1}\cdot\beta\cdot\alpha\), where \(\beta\) is a positive braid; the relations are \(((x_j\cdot \beta)\cdot x_j^{-1})\cdot \alpha\) where \(j\) is an integer of the `Tietze` word of \(\beta\). This presentation is not simplified by default since it represents the homotopy type of the complement of the curve.

    Note

    The curve must be defined over the rationals or a number field with an embedding over \(\QQbar\). This functionality requires the `sirocco` package to be installed.

    EXAMPLES:

    Sage

        sage: # needs sirocco
        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = A.curve(y^2 - x^3 - x^2)
        sage: C.fundamental_group(puiseux=False)
        Finitely presented group < x0 |  >
        sage: bm = C.braid_monodromy()
        sage: g = C.fundamental_group(simplified=False)
        sage: g.sorted_presentation()
        Finitely presented group < x0, x1 | x1^-1*x0^-1*x1*x0, x1^-1*x0 >
        sage: g.simplified()
        Finitely presented group < x0 |  >

    Python

        >>> from sage.all import *
        >>> # needs sirocco
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = A.curve(y**Integer(2) - x**Integer(3) - x**Integer(2))
        >>> C.fundamental_group(puiseux=False)
        Finitely presented group < x0 |  >
        >>> bm = C.braid_monodromy()
        >>> g = C.fundamental_group(simplified=False)
        >>> g.sorted_presentation()
        Finitely presented group < x0, x1 | x1^-1*x0^-1*x1*x0, x1^-1*x0 >
        >>> g.simplified()
        Finitely presented group < x0 |  >

    In the case of number fields, they need to have an embedding to the algebraic field:

    Sage

        sage: # needs sage.rings.number_field
        sage: x = polygen(ZZ)
        sage: a = QQ[x](x^2 + 5).roots(QQbar)[0][0]
        sage: F = NumberField(a.minpoly(), 'a', embedding=a)
        sage: F.inject_variables()
        Defining a
        sage: A.<x,y> = AffineSpace(F, 2)
        sage: C = A.curve(y^2 - a*x^3 - x^2)
        sage: C.fundamental_group()                     # needs sirocco
        Finitely presented group < x0 |  >
        sage: C = A.curve(x * (x - 1))
        sage: C.fundamental_group()                     # needs sirocco
        Finitely presented group < x0, x1 |  >

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> x = polygen(ZZ)
        >>> a = QQ[x](x**Integer(2) + Integer(5)).roots(QQbar)[Integer(0)][Integer(0)]
        >>> F = NumberField(a.minpoly(), 'a', embedding=a)
        >>> F.inject_variables()
        Defining a
        >>> A = AffineSpace(F, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = A.curve(y**Integer(2) - a*x**Integer(3) - x**Integer(2))
        >>> C.fundamental_group()                     # needs sirocco
        Finitely presented group < x0 |  >
        >>> C = A.curve(x * (x - Integer(1)))
        >>> C.fundamental_group()                     # needs sirocco
        Finitely presented group < x0, x1 |  >

  has_vertical_asymptote()

    Check if the curve is not a line and has vertical asymptotes.

    EXAMPLES:

    Sage

        sage: A2.<x,y> = AffineSpace(2, QQ)
        sage: Curve(x).has_vertical_asymptote()
        False
        sage: Curve(y^2 * x + x + y).has_vertical_asymptote()
        True

    Python

        >>> from sage.all import *
        >>> A2 = AffineSpace(Integer(2), QQ, names=('x', 'y',)); (x, y,) = A2._first_ngens(2)
        >>> Curve(x).has_vertical_asymptote()
        False
        >>> Curve(y**Integer(2) * x + x + y).has_vertical_asymptote()
        True

  is_vertical_line()

    Check if the curve is a vertical line.

    EXAMPLES:

    Sage

        sage: A2.<x, y> = AffineSpace(2, QQ)
        sage: Curve(x - 1).is_vertical_line()
        True
        sage: Curve(x - y).is_vertical_line()
        False
        sage: Curve(y^2 * x + x + y).is_vertical_line()
        False

    Python

        >>> from sage.all import *
        >>> A2 = AffineSpace(Integer(2), QQ, names=('x', 'y',)); (x, y,) = A2._first_ngens(2)
        >>> Curve(x - Integer(1)).is_vertical_line()
        True
        >>> Curve(x - y).is_vertical_line()
        False
        >>> Curve(y**Integer(2) * x + x + y).is_vertical_line()
        False

  riemann_surface(**kwargs)

    Return the complex Riemann surface determined by this curve.

    OUTPUT: a `RiemannSurface` object

    EXAMPLES:

    Sage

        sage: R.<x,y> = QQ[]
        sage: C = Curve(x^3 + 3*y^3 + 5)
        sage: C.riemann_surface()
        Riemann surface defined by polynomial f = x^3 + 3*y^3 + 5 = 0,
         with 53 bits of precision

    Python

        >>> from sage.all import *
        >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
        >>> C = Curve(x**Integer(3) + Integer(3)*y**Integer(3) + Integer(5))
        >>> C.riemann_surface()
        Riemann surface defined by polynomial f = x^3 + 3*y^3 + 5 = 0,
         with 53 bits of precision

class sage.schemes.curves.affine_curve.AffinePlaneCurve_finite_field(A, f)

  Bases: `AffinePlaneCurve_field`

  Affine plane curves over finite fields.

  rational_points(algorithm='enum')

    Return sorted list of all rational points on this curve.

    INPUT:

    - `algorithm` – possible choices:

      > - `'enum'` – use very naive point enumeration to find all rational points on this curve over a finite field.

      > - `'bn'` – via Singular’s Brill-Noether package.

      > - `'all'` – use all implemented algorithms and verify that they give the same answer, then return it

    Note

    The Brill-Noether package does not always work. When it fails, a `RuntimeError` exception is raised.

    EXAMPLES:

    Sage

        sage: x, y = (GF(5)['x,y']).gens()
        sage: f = y^2 - x^9 - x
        sage: C = Curve(f); C
        Affine Plane Curve over Finite Field of size 5 defined by -x^9 + y^2 - x
        sage: C.rational_points(algorithm='bn')
        [(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)]
        sage: C = Curve(x - y + 1)
        sage: C.rational_points()
        [(0, 1), (1, 2), (2, 3), (3, 4), (4, 0)]

    Python

        >>> from sage.all import *
        >>> x, y = (GF(Integer(5))['x,y']).gens()
        >>> f = y**Integer(2) - x**Integer(9) - x
        >>> C = Curve(f); C
        Affine Plane Curve over Finite Field of size 5 defined by -x^9 + y^2 - x
        >>> C.rational_points(algorithm='bn')
        [(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)]
        >>> C = Curve(x - y + Integer(1))
        >>> C.rational_points()
        [(0, 1), (1, 2), (2, 3), (3, 4), (4, 0)]

    We compare Brill-Noether and enumeration:

    Sage

        sage: x, y = (GF(17)['x,y']).gens()
        sage: C = Curve(x^2 + y^5 + x*y - 19)
        sage: v = C.rational_points(algorithm='bn')
        sage: w = C.rational_points(algorithm='enum')
        sage: len(v)
        20
        sage: v == w
        True

        sage: # needs sage.rings.finite_rings
        sage: A.<x,y> = AffineSpace(2, GF(9,'a'))
        sage: C = Curve(x^2 + y^2 - 1); C
        Affine Plane Curve over Finite Field in a of size 3^2
         defined by x^2 + y^2 - 1
        sage: C.rational_points()
        [(0, 1), (0, 2), (1, 0), (2, 0), (a + 1, a + 1),
         (a + 1, 2*a + 2), (2*a + 2, a + 1), (2*a + 2, 2*a + 2)]

    [Python]

        >>> from sage.all import *
        >>> x, y = (GF(Integer(17))['x,y']).gens()
        >>> C = Curve(x**Integer(2) + y**Integer(5) + x*y - Integer(19))
        >>> v = C.rational_points(algorithm='bn')
        >>> w = C.rational_points(algorithm='enum')
        >>> len(v)
        20
        >>> v == w
        True

        >>> # needs sage.rings.finite_rings
        >>> A = AffineSpace(Integer(2), GF(Integer(9),'a'), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(2) + y**Integer(2) - Integer(1)); C
        Affine Plane Curve over Finite Field in a of size 3^2
         defined by x^2 + y^2 - 1
        >>> C.rational_points()
        [(0, 1), (0, 2), (1, 0), (2, 0), (a + 1, a + 1),
         (a + 1, 2*a + 2), (2*a + 2, a + 1), (2*a + 2, 2*a + 2)]

  riemann_roch_basis(D)

    Return a basis of the Riemann-Roch space of the divisor `D`.

    This interfaces with Singular’s Brill-Noether command.

    This curve is assumed to be a plane curve defined by a polynomial equation \(f(x,y) = 0\) over a prime finite field \(F = GF(p)\) in 2 variables \(x,y\) representing a curve \(X: f(x,y) = 0\) having \(n\) \(F\)-rational points (see the Sage function `places_on_curve`)

    INPUT:

    - `D` – an \(n\)-tuple of integers \((d_1, ..., d_n)\) representing the divisor \(Div = d_1P_1 + \dots + d_nP_n\), where \(X(F) = \{P_1, \dots, P_n\}\). The ordering is that dictated by `places_on_curve`.

    OUTPUT: a basis of \(L(Div)\).

    EXAMPLES:

    Sage

        sage: R = PolynomialRing(GF(5), 2, names=["x","y"])
        sage: x, y = R.gens()
        sage: f = y^2 - x^9 - x
        sage: C = Curve(f)
        sage: D = [6,0,0,0,0,0]
        sage: C.riemann_roch_basis(D)
        [1, (-x*z^5 + y^2*z^4)/x^6, (-x*z^6 + y^2*z^5)/x^7, (-x*z^7 + y^2*z^6)/x^8]

    Python

        >>> from sage.all import *
        >>> R = PolynomialRing(GF(Integer(5)), Integer(2), names=["x","y"])
        >>> x, y = R.gens()
        >>> f = y**Integer(2) - x**Integer(9) - x
        >>> C = Curve(f)
        >>> D = [Integer(6),Integer(0),Integer(0),Integer(0),Integer(0),Integer(0)]
        >>> C.riemann_roch_basis(D)
        [1, (-x*z^5 + y^2*z^4)/x^6, (-x*z^6 + y^2*z^5)/x^7, (-x*z^7 + y^2*z^6)/x^8]

class sage.schemes.curves.affine_curve.IntegralAffineCurve(A, X)

  Bases: `AffineCurve_field`

  Base class for integral affine curves.

  coordinate_functions()

    Return the coordinate functions.

    EXAMPLES:

    Sage

        sage: # needs sage.rings.finite_rings
        sage: A.<x,y> = AffineSpace(GF(8), 2)
        sage: C = Curve(x^5 + y^5 + x*y + 1)
        sage: x, y = C.coordinate_functions()
        sage: x^5 + y^5 + x*y + 1
        0

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.finite_rings
        >>> A = AffineSpace(GF(Integer(8)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(5) + y**Integer(5) + x*y + Integer(1))
        >>> x, y = C.coordinate_functions()
        >>> x**Integer(5) + y**Integer(5) + x*y + Integer(1)
        0

  function(f)

    Return the function field element coerced from `f`.

    INPUT:

    - `f` – an element of the fraction field of the coordinate ring of the ambient space or the coordinate ring of the curve

    OUTPUT: an element of the function field of this curve

    EXAMPLES:

    Sage

        sage: # needs sage.rings.finite_rings
        sage: A.<x,y> = AffineSpace(GF(8), 2)
        sage: C = Curve(x^5 + y^5 + x*y + 1)
        sage: f = C.function(x/y)
        sage: f
        (x/(x^5 + 1))*y^4 + x^2/(x^5 + 1)
        sage: df = f.differential(); df
        ((1/(x^10 + 1))*y^4 + x^6/(x^10 + 1)) d(x)
        sage: df.divisor()                  # long time
        2*Place (1/x, 1/x^4*y^4 + 1/x^3*y^3 + 1/x^2*y^2 + 1/x*y + 1)
         + 2*Place (1/x, 1/x*y + 1)
         - 2*Place (x + 1, y)
         - 2*Place (x^4 + x^3 + x^2 + x + 1, y)

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.finite_rings
        >>> A = AffineSpace(GF(Integer(8)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(5) + y**Integer(5) + x*y + Integer(1))
        >>> f = C.function(x/y)
        >>> f
        (x/(x^5 + 1))*y^4 + x^2/(x^5 + 1)
        >>> df = f.differential(); df
        ((1/(x^10 + 1))*y^4 + x^6/(x^10 + 1)) d(x)
        >>> df.divisor()                  # long time
        2*Place (1/x, 1/x^4*y^4 + 1/x^3*y^3 + 1/x^2*y^2 + 1/x*y + 1)
         + 2*Place (1/x, 1/x*y + 1)
         - 2*Place (x + 1, y)
         - 2*Place (x^4 + x^3 + x^2 + x + 1, y)

  function_field()

    Return the function field of the curve.

    EXAMPLES:

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve(x^3 - y^2 - x^4 - y^4)
        sage: C.function_field()
        Function field in y defined by y^4 + y^2 + x^4 - x^3

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(3) - y**Integer(2) - x**Integer(4) - y**Integer(4))
        >>> C.function_field()
        Function field in y defined by y^4 + y^2 + x^4 - x^3

    Sage

        sage: # needs sage.rings.finite_rings
        sage: A.<x,y> = AffineSpace(GF(8), 2)
        sage: C = Curve(x^5 + y^5 + x*y + 1)
        sage: C.function_field()
        Function field in y defined by y^5 + x*y + x^5 + 1

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.finite_rings
        >>> A = AffineSpace(GF(Integer(8)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(5) + y**Integer(5) + x*y + Integer(1))
        >>> C.function_field()
        Function field in y defined by y^5 + x*y + x^5 + 1

  parametric_representation(place, name=None)

    Return a power series representation of the branch of the curve given by `place`.

    INPUT:

    - `place` – a place on the curve

    EXAMPLES:

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve(x^2 + y^2 -1)
        sage: p = C(0,1)
        sage: p.closed_point()
        Point (x, y - 1)
        sage: pl = _.place()
        sage: C.parametric_representation(pl)
        (s + ..., 1 - 1/2*s^2 - 1/8*s^4 - 1/16*s^6 + ...)

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(2) + y**Integer(2) -Integer(1))
        >>> p = C(Integer(0),Integer(1))
        >>> p.closed_point()
        Point (x, y - 1)
        >>> pl = _.place()
        >>> C.parametric_representation(pl)
        (s + ..., 1 - 1/2*s^2 - 1/8*s^4 - 1/16*s^6 + ...)

    Sage

        sage: # needs sage.rings.finite_rings
        sage: A.<x,y> = AffineSpace(GF(7^2), 2)
        sage: C = Curve(x^2 - x^4 - y^4)
        sage: p, = C.singular_closed_points()
        sage: b1, b2 = p.places()
        sage: xs, ys = C.parametric_representation(b1)
        sage: f = xs^2 - xs^4 - ys^4
        sage: [f.coefficient(i) for i in range(5)]
        [0, 0, 0, 0, 0]
        sage: xs, ys = C.parametric_representation(b2)
        sage: f = xs^2 - xs^4 - ys^4
        sage: [f.coefficient(i) for i in range(5)]
        [0, 0, 0, 0, 0]

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.finite_rings
        >>> A = AffineSpace(GF(Integer(7)**Integer(2)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(2) - x**Integer(4) - y**Integer(4))
        >>> p, = C.singular_closed_points()
        >>> b1, b2 = p.places()
        >>> xs, ys = C.parametric_representation(b1)
        >>> f = xs**Integer(2) - xs**Integer(4) - ys**Integer(4)
        >>> [f.coefficient(i) for i in range(Integer(5))]
        [0, 0, 0, 0, 0]
        >>> xs, ys = C.parametric_representation(b2)
        >>> f = xs**Integer(2) - xs**Integer(4) - ys**Integer(4)
        >>> [f.coefficient(i) for i in range(Integer(5))]
        [0, 0, 0, 0, 0]

  place_to_closed_point(place)

    Return the closed point on the place.

    INPUT:

    - `place` – a place of the function field of the curve

    EXAMPLES:

    Sage

        sage: # needs sage.rings.finite_rings
        sage: A.<x,y> = AffineSpace(GF(4), 2)
        sage: C = Curve(x^5 + y^5 + x*y + 1)
        sage: F = C.function_field()
        sage: pls = F.places(1)
        sage: C.place_to_closed_point(pls[-1])
        Point (x + 1, y + 1)
        sage: C.place_to_closed_point(pls[-2])
        Point (x + 1, y + 1)

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.finite_rings
        >>> A = AffineSpace(GF(Integer(4)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(5) + y**Integer(5) + x*y + Integer(1))
        >>> F = C.function_field()
        >>> pls = F.places(Integer(1))
        >>> C.place_to_closed_point(pls[-Integer(1)])
        Point (x + 1, y + 1)
        >>> C.place_to_closed_point(pls[-Integer(2)])
        Point (x + 1, y + 1)

  places_at_infinity()

    Return the places of the curve at infinity.

    EXAMPLES:

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve(x^3 - y^2 - x^4 - y^4)
        sage: C.places_at_infinity()
        [Place (1/x, 1/x^2*y, 1/x^3*y^2, 1/x^4*y^3)]

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(3) - y**Integer(2) - x**Integer(4) - y**Integer(4))
        >>> C.places_at_infinity()
        [Place (1/x, 1/x^2*y, 1/x^3*y^2, 1/x^4*y^3)]

    Sage

        sage: # needs sage.rings.finite_rings
        sage: F = GF(9)
        sage: A2.<x,y> = AffineSpace(F, 2)
        sage: C = A2.curve(y^3 + y - x^4)
        sage: C.places_at_infinity()
        [Place (1/x, 1/x^3*y^2)]

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.finite_rings
        >>> F = GF(Integer(9))
        >>> A2 = AffineSpace(F, Integer(2), names=('x', 'y',)); (x, y,) = A2._first_ngens(2)
        >>> C = A2.curve(y**Integer(3) + y - x**Integer(4))
        >>> C.places_at_infinity()
        [Place (1/x, 1/x^3*y^2)]

    Sage

        sage: A.<x,y,z> = AffineSpace(GF(11), 3)
        sage: C = Curve([x*z - y^2, y - z^2, x - y*z], A)
        sage: C.places_at_infinity()
        [Place (1/x, 1/x*z^2)]

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(GF(Integer(11)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
        >>> C = Curve([x*z - y**Integer(2), y - z**Integer(2), x - y*z], A)
        >>> C.places_at_infinity()
        [Place (1/x, 1/x*z^2)]

  places_on(point)

    Return the places on the closed point.

    INPUT:

    - `point` – a closed point of the curve

    OUTPUT: list of the places of the function field of the curve

    EXAMPLES:

    Sage

        sage: A.<x,y> = AffineSpace(QQ, 2)
        sage: C = Curve(x^3 - y^2 - x^4 - y^4)
        sage: C.singular_closed_points()
        [Point (x, y)]
        sage: p, = _
        sage: C.places_on(p)
        [Place (x, y, y^2, 1/x*y^3 + 1/x*y)]

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(3) - y**Integer(2) - x**Integer(4) - y**Integer(4))
        >>> C.singular_closed_points()
        [Point (x, y)]
        >>> p, = _
        >>> C.places_on(p)
        [Place (x, y, y^2, 1/x*y^3 + 1/x*y)]

    Sage

        sage: # needs sage.rings.finite_rings
        sage: k.<a> = GF(9)
        sage: A.<x,y> = AffineSpace(k, 2)
        sage: C = Curve(y^2 - x^5 - x^4 - 2*x^3 - 2*x - 2)
        sage: pts = C.closed_points()
        sage: pts
        [Point (x, y + (a + 1)),
         Point (x, y + (-a - 1)),
         Point (x + (a + 1), y + (a - 1)),
         Point (x + (a + 1), y + (-a + 1)),
         Point (x - 1, y + (a + 1)),
         Point (x - 1, y + (-a - 1)),
         Point (x + (-a - 1), y + a),
         Point (x + (-a - 1), y + (-a)),
         Point (x + 1, y + 1),
         Point (x + 1, y - 1)]
        sage: p1, p2, p3 = pts[:3]
        sage: C.places_on(p1)
        [Place (x, y + a + 1)]
        sage: C.places_on(p2)
        [Place (x, y + 2*a + 2)]
        sage: C.places_on(p3)
        [Place (x + a + 1, y + a + 2)]

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.finite_rings
        >>> k = GF(Integer(9), names=('a',)); (a,) = k._first_ngens(1)
        >>> A = AffineSpace(k, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(y**Integer(2) - x**Integer(5) - x**Integer(4) - Integer(2)*x**Integer(3) - Integer(2)*x - Integer(2))
        >>> pts = C.closed_points()
        >>> pts
        [Point (x, y + (a + 1)),
         Point (x, y + (-a - 1)),
         Point (x + (a + 1), y + (a - 1)),
         Point (x + (a + 1), y + (-a + 1)),
         Point (x - 1, y + (a + 1)),
         Point (x - 1, y + (-a - 1)),
         Point (x + (-a - 1), y + a),
         Point (x + (-a - 1), y + (-a)),
         Point (x + 1, y + 1),
         Point (x + 1, y - 1)]
        >>> p1, p2, p3 = pts[:Integer(3)]
        >>> C.places_on(p1)
        [Place (x, y + a + 1)]
        >>> C.places_on(p2)
        [Place (x, y + 2*a + 2)]
        >>> C.places_on(p3)
        [Place (x + a + 1, y + a + 2)]

    Sage

        sage: # needs sage.rings.finite_rings
        sage: F.<a> = GF(8)
        sage: P.<x,y,z> = ProjectiveSpace(F, 2)
        sage: Cp = Curve(x^3*y + y^3*z + x*z^3)
        sage: C = Cp.affine_patch(0)

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.finite_rings
        >>> F = GF(Integer(8), names=('a',)); (a,) = F._first_ngens(1)
        >>> P = ProjectiveSpace(F, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
        >>> Cp = Curve(x**Integer(3)*y + y**Integer(3)*z + x*z**Integer(3))
        >>> C = Cp.affine_patch(Integer(0))

  pull_from_function_field(f)

    Return the fraction corresponding to `f`.

    INPUT:

    - `f` – an element of the function field

    OUTPUT:

    A fraction of polynomials in the coordinate ring of the ambient space of the curve.

    EXAMPLES:

    Sage

        sage: # needs sage.rings.finite_rings
        sage: A.<x,y> = AffineSpace(GF(8), 2)
        sage: C = Curve(x^5 + y^5 + x*y + 1)
        sage: F = C.function_field()
        sage: C.pull_from_function_field(F.gen())
        y
        sage: C.pull_from_function_field(F.one())
        1
        sage: C.pull_from_function_field(F.zero())
        0
        sage: f1 = F.gen()
        sage: f2 = F.base_ring().gen()
        sage: C.function(C.pull_from_function_field(f1)) == f1
        True
        sage: C.function(C.pull_from_function_field(f2)) == f2
        True

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.finite_rings
        >>> A = AffineSpace(GF(Integer(8)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(5) + y**Integer(5) + x*y + Integer(1))
        >>> F = C.function_field()
        >>> C.pull_from_function_field(F.gen())
        y
        >>> C.pull_from_function_field(F.one())
        1
        >>> C.pull_from_function_field(F.zero())
        0
        >>> f1 = F.gen()
        >>> f2 = F.base_ring().gen()
        >>> C.function(C.pull_from_function_field(f1)) == f1
        True
        >>> C.function(C.pull_from_function_field(f2)) == f2
        True

  singular_closed_points()

    Return the singular closed points of the curve.

    EXAMPLES:

    Sage

        sage: # needs sage.rings.finite_rings
        sage: A.<x,y> = AffineSpace(GF(7^2), 2)
        sage: C = Curve(x^2 - x^4 - y^4)
        sage: C.singular_closed_points()
        [Point (x, y)]

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.finite_rings
        >>> A = AffineSpace(GF(Integer(7)**Integer(2)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(2) - x**Integer(4) - y**Integer(4))
        >>> C.singular_closed_points()
        [Point (x, y)]

    Sage

        sage: A.<x,y,z> = AffineSpace(GF(11), 3)
        sage: C = Curve([x*z - y^2, y - z^2, x - y*z], A)
        sage: C.singular_closed_points()
        []

    [Python]

        >>> from sage.all import *
        >>> A = AffineSpace(GF(Integer(11)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
        >>> C = Curve([x*z - y**Integer(2), y - z**Integer(2), x - y*z], A)
        >>> C.singular_closed_points()
        []

class sage.schemes.curves.affine_curve.IntegralAffineCurve_finite_field(A, X)

  Bases: `IntegralAffineCurve`

  Integral affine curves.

  INPUT:

  - `A` – an ambient space in which the curve lives

  - `X` – list of polynomials that define the curve

  EXAMPLES:

  Sage

      sage: A.<x,y,z> = AffineSpace(GF(11), 3)
      sage: C = Curve([x*z - y^2, y - z^2, x - y*z], A); C
      Affine Curve over Finite Field of size 11
       defined by -y^2 + x*z, -z^2 + y, -y*z + x
      sage: C.function_field()
      Function field in z defined by z^3 + 10*x

  Python

      >>> from sage.all import *
      >>> A = AffineSpace(GF(Integer(11)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
      >>> C = Curve([x*z - y**Integer(2), y - z**Integer(2), x - y*z], A); C
      Affine Curve over Finite Field of size 11
       defined by -y^2 + x*z, -z^2 + y, -y*z + x
      >>> C.function_field()
      Function field in z defined by z^3 + 10*x

  closed_points(degree=1)

    Return a list of the closed points of `degree` of the curve.

    INPUT:

    - `degree` – positive integer

    EXAMPLES:

    Sage

        sage: A.<x,y> = AffineSpace(GF(7), 2)
        sage: C = Curve(x^2 - x^4 - y^4)
        sage: C.closed_points()
        [Point (x, y),
         Point (x + 1, y),
         Point (x + 2, y + 2),
         Point (x + 2, y - 2),
         Point (x - 2, y + 2),
         Point (x - 2, y - 2),
         Point (x - 1, y)]

    Python

        >>> from sage.all import *
        >>> A = AffineSpace(GF(Integer(7)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
        >>> C = Curve(x**Integer(2) - x**Integer(4) - y**Integer(4))
        >>> C.closed_points()
        [Point (x, y),
         Point (x + 1, y),
         Point (x + 2, y + 2),
         Point (x + 2, y - 2),
         Point (x - 2, y + 2),
         Point (x - 2, y - 2),
         Point (x - 1, y)]

  places(degree=1)

    Return all places on the curve of the `degree`.

    INPUT:

    - `degree` – positive integer

    EXAMPLES:

    Sage

        sage: # needs sage.rings.finite_rings
        sage: F = GF(9)
        sage: A2.<x,y> = AffineSpace(F, 2)
        sage: C = A2.curve(y^3 + y - x^4)
        sage: C.places()
        [Place (1/x, 1/x^3*y^2),
         Place (x, y),
         Place (x, y + z2 + 1),
         Place (x, y + 2*z2 + 2),
         Place (x + z2, y + 2),
         Place (x + z2, y + z2),
         Place (x + z2, y + 2*z2 + 1),
         Place (x + z2 + 1, y + 1),
         Place (x + z2 + 1, y + z2 + 2),
         Place (x + z2 + 1, y + 2*z2),
         Place (x + 2*z2 + 1, y + 2),
         Place (x + 2*z2 + 1, y + z2),
         Place (x + 2*z2 + 1, y + 2*z2 + 1),
         Place (x + 2, y + 1),
         Place (x + 2, y + z2 + 2),
         Place (x + 2, y + 2*z2),
         Place (x + 2*z2, y + 2),
         Place (x + 2*z2, y + z2),
         Place (x + 2*z2, y + 2*z2 + 1),
         Place (x + 2*z2 + 2, y + 1),
         Place (x + 2*z2 + 2, y + z2 + 2),
         Place (x + 2*z2 + 2, y + 2*z2),
         Place (x + z2 + 2, y + 2),
         Place (x + z2 + 2, y + z2),
         Place (x + z2 + 2, y + 2*z2 + 1),
         Place (x + 1, y + 1),
         Place (x + 1, y + z2 + 2),
         Place (x + 1, y + 2*z2)]

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.finite_rings
        >>> F = GF(Integer(9))
        >>> A2 = AffineSpace(F, Integer(2), names=('x', 'y',)); (x, y,) = A2._first_ngens(2)
        >>> C = A2.curve(y**Integer(3) + y - x**Integer(4))
        >>> C.places()
        [Place (1/x, 1/x^3*y^2),
         Place (x, y),
         Place (x, y + z2 + 1),
         Place (x, y + 2*z2 + 2),
         Place (x + z2, y + 2),
         Place (x + z2, y + z2),
         Place (x + z2, y + 2*z2 + 1),
         Place (x + z2 + 1, y + 1),
         Place (x + z2 + 1, y + z2 + 2),
         Place (x + z2 + 1, y + 2*z2),
         Place (x + 2*z2 + 1, y + 2),
         Place (x + 2*z2 + 1, y + z2),
         Place (x + 2*z2 + 1, y + 2*z2 + 1),
         Place (x + 2, y + 1),
         Place (x + 2, y + z2 + 2),
         Place (x + 2, y + 2*z2),
         Place (x + 2*z2, y + 2),
         Place (x + 2*z2, y + z2),
         Place (x + 2*z2, y + 2*z2 + 1),
         Place (x + 2*z2 + 2, y + 1),
         Place (x + 2*z2 + 2, y + z2 + 2),
         Place (x + 2*z2 + 2, y + 2*z2),
         Place (x + z2 + 2, y + 2),
         Place (x + z2 + 2, y + z2),
         Place (x + z2 + 2, y + 2*z2 + 1),
         Place (x + 1, y + 1),
         Place (x + 1, y + z2 + 2),
         Place (x + 1, y + 2*z2)]

class sage.schemes.curves.affine_curve.IntegralAffinePlaneCurve(A, f)

  Bases: `IntegralAffineCurve`, `AffinePlaneCurve_field`

class sage.schemes.curves.affine_curve.IntegralAffinePlaneCurve_finite_field(A, f)

  Bases: `AffinePlaneCurve_finite_field`, `IntegralAffineCurve_finite_field`

  Integral affine plane curve over a finite field.

  EXAMPLES:

  Sage

      sage: # needs sage.rings.finite_rings
      sage: A.<x,y> = AffineSpace(GF(8), 2)
      sage: C = Curve(x^5 + y^5 + x*y + 1); C
      Affine Plane Curve over Finite Field in z3 of size 2^3
       defined by x^5 + y^5 + x*y + 1
      sage: C.function_field()
      Function field in y defined by y^5 + x*y + x^5 + 1

  Python

      >>> from sage.all import *
      >>> # needs sage.rings.finite_rings
      >>> A = AffineSpace(GF(Integer(8)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
      >>> C = Curve(x**Integer(5) + y**Integer(5) + x*y + Integer(1)); C
      Affine Plane Curve over Finite Field in z3 of size 2^3
       defined by x^5 + y^5 + x*y + 1
      >>> C.function_field()
      Function field in y defined by y^5 + x*y + x^5 + 1