Singular Manual: puiseuxExpansion

| #### D.13.6.3 puiseuxExpansion Procedure from library `tropical.lib` (see tropical_lib). Usage: puiseuxExpansion(f,n,#); f poly, n int, # list Assume: f is a non-constant polynomial in two variables which is not divisible by the first variable and which is squarefree as a power series over the complex numbers; the base field is either the field of rational numbers or a finite extension thereof; monomial ordering is assumed to be local; the optional parameter # can be the string 'subst' Return: list, where each entry of the list l describes the Newton-Puiseux parametrisations of one branch of the plane curve singularity at the origin defined by f; only the terms up to order n of each parametetrisation are computed l[i][1] = is a ring l[i][2] = int l[i][3] = string WE NOW DESCRIBE THE LIST ENTRIES l[i] IN MORE DETAIL: - the ring l[i][1] contains an ideal LIFT and the Newton-Puiseux parametrisation of the branch is given by x=t^N and y=LIFT[1], where N=l[i][2] - if the base field had a parameter and a minimal polynomial, then the new base field will have a parameter and a new minimal polynomial, and LIFT[2] describes how the old parameter can be computed from the new one - if a field extension with minimal polynomial of degree k was necessary, then to the one extension produced actually k extensions correspond by replacing the parameter a successively by all zeros of the minimal polynomial - if the option subst was set l[i][3] contains the polynomial where y has been substituted by y(t^{1/N}) as a string Remark: - it is best to use the procedure displayPuiseuxExpansion to display the result - the procedure requires the Singular procedure absPrimdecGTZ to be present in the package primdec.lib - if f is not squarefree it will be replaced by its squarefree part Example:
    LIB "tropical.lib";
    printlevel=1;
    ring r=0,(x,y),ds;
    poly f=x2-y4+x5y7;
    puiseuxExpansion(f,3,"subst");
    displayPuiseuxExpansion(puiseuxExpansion(f,3));

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