Singular Manual: vandermonde

  | ### 5.1.164 vandermonde `Syntax:` `vandermonde (` ideal_expression`,` ideal_expression`,` int_expression `)` `Type:` poly `Purpose:` vandermonde(p,v,d) computes the (unique) polynomial of degree d with prescribed values v[1],...,v[N] at the points p182#182 p183#183, N=(d+1)184#184, 17#17 the number of ring variables. The returned polynomial is 185#185, where the coefficients 186#186 are the solution of the (transposed) Vandermonde system of linear equations 187#187 `Note:` the ground field has to be the field of rational numbers. Moreover, ncols(p)==17#17, the number of variables in the basering, and all the given generators have to be numbers different from 0,1 or -1. Finally, ncols(v)==(d+1)184#184, and all given generators have to be numbers. `Example:`
      ring r=0,(x,y),dp;
      // determine f with deg(f)=2 and with given values v of f
      // at 9 points: (2,3)^0=(1,1),...,(2,3)^8=(2^8,3^8)
      // valuation point: (2,3)
      ideal p=2,3;
      ideal v=1,2,3,4,5,6,7,8,9;
      poly ip=vandermonde(p,v,2);
      ip[1..5];  //  the 5 first terms of ip:
      ==> -1/9797760x2y2-595/85536x2y+55/396576xy2+935/384x2-1309/3240xy
      // compute value of ip at the point 2^8,3^8, result must be 9
      subst(subst(ip,x,2^8),y,3^8);
      ==> 9

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