### 5.1.63 interpolation

`Syntax:`
  `interpolation (` list`,` intvec `)`
`Type:`
  ideal
`Purpose:`
  `interpolation(l,v)` computes the reduced Groebner basis of the intersection of ideals l[1]^v[1], ..., l[N]^v[N] by applying linear algebra methods.
`Assume:`
  Every ideal from the list l must be a maximal ideal of a point and should have the following form: variable_1-coordinate_1, ..., variable_n-coordinate_n, where n is the number of variables in the ring.
  The ring should be a polynomial ring over Zp or Q with global ordering.
`Example:`

        ring r=0,(x,y),dp;
        ideal p_1=x,y;
        ideal p_2=x+1,y+1;
        ideal p_3=x+2,y-1;
        ideal p_4=x-1,y+2;
        ideal p_5=x-1,y-3;
        ideal p_6=x,y+3;
        ideal p_7=x+2,y;
        list l=p_1,p_2,p_3,p_4,p_5,p_6,p_7;
        intvec v=2,1,1,1,1,1,1;
        ideal j=interpolation(l,v);
        // generator of degree 3 gives the equation of the unique
        // singular cubic passing
        // through p_1,...,p_7 with singularity at p_1
        j;
      ==> j[1]=-4x3-4x2y-2xy2+y3-8x2-4xy+3y2
      ==> j[2]=-y4+8x2y+6xy2-2y3+10xy+3y2
      ==> j[3]=-xy3+2x2y+xy2+4xy
      ==> j[4]=-2x2y2-2x2y-2xy2+y3-4xy+3y2
        // computes values of generators of j at p_4, results should be 0
        subst(j,x,1,y,-2);
      ==> _[1]=0
      ==> _[2]=0
      ==> _[3]=0
      ==> _[4]=0
        // computes values of derivatives d/dx of generators at (0,0)
        subst(diff(j,x),x,0,y,0);
      ==> _[1]=0
      ==> _[2]=0
      ==> _[3]=0
      ==> _[4]=0

See diff; fglm; intersect; std; subst.

User manual for Singular version 4.4.1, 2025, generated by texi2html.