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A.3.12 Algebraic dependence

Let $g$, $f_1$, ..., $f_r\in K[x_1,\ldots,x_n]$.We want to check whether

  1. $f_1$, ..., $f_r$are algebraically dependent.

    Let $I=\langle Y_1-f_1,\ldots,Y_r-f_r \rangle \subseteq K[x_1,\ldots,x_n,Y_1,\ldots,Y_r]$.Then $I \cap K[Y_1,\ldots,Y_r]$are the algebraic relations between $f_1$, ..., $f_r$.

  2. $g \in K [f_1,\ldots,f_r]$.

    $g \in K [f_1,\ldots,f_r]$if and only if the normal form of $g$ with respect to $I$ and a block ordering with respect to $X=(x_1,\ldots,x_n)$ and $Y=(Y_1,\ldots,Y_r)$ with $X>Y$is in $K[Y]$.

Both questions can be answered using the following procedure. If the second argument is zero, it checks for algebraic dependence and returns the ideal of relations between the generators of the given ideal. Otherwise it checks for subring membership and returns the normal form of the second argument with respect to the ideal I.

 
  proc algebraicDep(ideal J, poly g)
  {
    def R=basering;         // give a name to the basering
    int n=size(J);
    int k=nvars(R);
    int i;
    intvec v;

    // construction of the new ring:

    // construct a weight vector
    v[n+k]=0;         // gives a zero vector of length n+k
    for(i=1;i<=k;i++)
    {
      v[i]=1;
    }
    string orde="(a("+string(v)+"),dp);";
    string ri="ring Rhelp=("+charstr(R)+"),
                          ("+varstr(R)+",Y(1.."+string(n)+")),"+orde;
                            // ring definition as a string
    execute(ri);            // execution of the string

    // construction of the new ideal I=(J[1]-Y(1),...,J[n]-Y(n))
    ideal I=imap(R,J);
    for(i=1;i<=n;i++)
    {
      I[i]=I[i]-var(k+i);
    }
    poly g=imap(R,g);
    if(g==0)
    {
      // construction of the ideal of relations by elimination
      poly el=var(1);
      for(i=2;i<=k;i++)
      {
        el=el*var(i);
      }
      ideal KK=eliminate(I,el);
      keepring(Rhelp);
      return(KK);
    }
    // reduction of g with respect to I
    ideal KK=reduce(g,std(I));
    keepring(Rhelp);
    return(KK);
  }

  // applications of the procedure
  ring r=0,(x,y,z),dp;
  ideal i=xz,yz;
  algebraicDep(i,0);
==> _[1]=0
  // Note: after call of algebraicDep(), the basering is Rhelp.
  setring r; kill Rhelp;
  ideal j=xy+z2,z2+y2,x2y2-2xy3+y4;
  algebraicDep(j,0);
==> _[1]=Y(1)^2-2*Y(1)*Y(2)+Y(2)^2-Y(3)
  setring r; kill Rhelp;
  poly g=y2z2-xz;
  algebraicDep(i,g);
==> _[1]=Y(2)^2-Y(1)
  // this shows that g is contained in i.
  setring r; kill Rhelp;
  algebraicDep(j,g);
==> _[1]=-z^4+z^2*Y(2)-x*z
  // this shows that g is contained in j.