### 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 `p` `p`, `N=(d+1)`, the number of ring variables.

  The returned polynomial is , where the coefficients are the solution of the (transposed) Vandermonde system of linear equations

`Note:`
  the ground field has to be the field of rational numbers. Moreover, `ncols(p)==`, 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)`, 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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