### A.3.2 Finite fields

We define a variety in the -space of codimension 2 defined by polynomials of degree with generic coefficients over the prime field and look for zeros on the torus. First over the prime field and then in the finite extension field with elements. In general there will be many more solutions in the second case. (Since the SINGULAR language is interpreted, the evaluation of many `for`-loops is not very fast):

      int p=3;  int n=3;  int d=5; int k=2;
      ring rp = p,(x(1..n)),dp;
      int s = size(maxideal(d));
      s;
    ==> 21
      // create a dense homogeneous ideal m, all generators of degree d, with
      // generic (random) coefficients:
      ideal m = maxideal(d)*random(p,s,n-2);
      m;
    ==> m[1]=x(1)^3*x(2)^2-x(1)*x(2)^4+x(1)^4*x(3)-x(1)^3*x(2)*x(3)+x(1)*x(2)^3*x\
       (3)+x(2)^4*x(3)+x(2)^3*x(3)^2+x(1)*x(2)*x(3)^3+x(1)*x(3)^4-x(3)^5
      // look for zeros on the torus by checking all points (with no component 0)
      // of the affine n-space over the field with p elements :
      ideal mt;
      int i(1..n);                    // initialize integers i(1),...,i(n)
      int l;
      s=0;
      for (i(1)=1;i(1)<p;i(1)=i(1)+1)
      {
        for (i(2)=1;i(2)<p;i(2)=i(2)+1)
        {
          for (i(3)=1;i(3)<p;i(3)=i(3)+1)
          {
            mt=m;
            for (l=1;l<=n;l=l+1)
            {
              mt=subst(mt,x(l),i(l));
            }
            if (size(mt)==0)
            {
              "solution:",i(1..n);
              s=s+1;
            }
          }
        }
      }
    ==> solution: 1 1 2
    ==> solution: 1 2 1
    ==> solution: 1 2 2
    ==> solution: 2 1 1
    ==> solution: 2 1 2
    ==> solution: 2 2 1
      "//",s,"solutions over GF("+string(p)+")";
    ==> // 6 solutions over GF(3)
      // Now go to the field with p^3 elements:
      // As long as there is no map from Z/p to the field with p^3 elements
      // implemented, use the following trick: convert the ideal to be mapped
      // to the new ring to a string and then execute this string in the
      // new ring
      string ms="ideal m="+string(m)+";";
      ms;
    ==> ideal m=x(1)^3*x(2)^2-x(1)*x(2)^4+x(1)^4*x(3)-x(1)^3*x(2)*x(3)+x(1)*x(2)^\
       3*x(3)+x(2)^4*x(3)+x(2)^3*x(3)^2+x(1)*x(2)*x(3)^3+x(1)*x(3)^4-x(3)^5;
      // define a ring rpk with p^k elements, call the primitive element z. Hence
      // 'solution exponent: 0 1 5' means that (z^0,z^1,z^5) is a solution
      ring rpk=(p^k,z),(x(1..n)),dp;
      rpk;
    ==> // coefficients: ZZ/9[z]
    ==> //   minpoly        : 1*z^2+2*z^1+2*z^0 considered as a field
    ==> // number of vars : 3
    ==> //        block   1 : ordering dp
    ==> //                  : names    x(1) x(2) x(3)
    ==> //        block   2 : ordering C
      execute(ms);
      s=0;
      ideal mt;
      for (i(1)=0;i(1)<p^k-1;i(1)=i(1)+1)
      {
        for (i(2)=0;i(2)<p^k-1;i(2)=i(2)+1)
        {
          for (i(3)=0;i(3)<p^k-1;i(3)=i(3)+1)
          {
            mt=m;
            for (l=1;l<=n;l=l+1)
            {
              mt=subst(mt,x(l),z^i(l));
            }
            if (size(mt)==0)
            {
              // we show only the first 7 solutions here:
              if (s<5) {"solution exponent:",i(1..n);}
              s=s+1;
            }
          }
        }
      }
    ==> solution exponent: 0 0 2
    ==> solution exponent: 0 0 4
    ==> solution exponent: 0 0 6
    ==> solution exponent: 0 4 0
    ==> solution exponent: 0 4 1
      "//",s,"solutions over GF("+string(p^k)+")";
    ==> // 72 solutions over GF(9)

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