### A.1.7 Cyclic roots

We write a procedure returning a string that enables us to create automatically the ideal of cyclic roots over the basering with n variables. The procedure assumes that the variables consist of a single letter each (hence no indexed variables are allowed; the procedure `cyclic` in `polylib.lib` does not have this restriction). Then we compute a standard basis of this ideal and some numerical information. (This ideal is used as a classical benchmark for standard basis computations).

    // We call the procedure 'cyclic':
    proc cyclic (int n)
    {
       string vs = varstr(basering)+varstr(basering);
       int c=find(vs,",");
       while ( c!=0 )
       {
          vs=vs[1,c-1]+vs[c+1,size(vs)];
          c=find(vs,",");
       }
       string t,s;
       int i,j;
       for ( j=1; j<=n-1; j=j+1 )
       {
          t="";
          for ( i=1; i <=n; i=i+1 )
          {
             t = t + vs[i,j] + "+";
          }
          t = t[1,size(t)-1] + ","+newline;
          s=s+t;
       }
       s=s+vs[1,n]+"-1";
       return (s);
    }

    ring r=0,(a,b,c,d,e),lp;         // basering, char 0, lex ordering
    string sc=cyclic(nvars(basering));
    sc;                              // the string of the ideal
    ==> a+b+c+d+e,
    ==> ab+bc+cd+de+ea,
    ==> abc+bcd+cde+dea+eab,
    ==> abcd+bcde+cdea+deab+eabc,
    ==> abcde-1
    execute("ideal i="+sc+";");      // this defines the ideal of cyclic roots
    i;
    ==> i[1]=a+b+c+d+e
    ==> i[2]=ab+bc+cd+ae+de
    ==> i[3]=abc+bcd+abe+ade+cde
    ==> i[4]=abcd+abce+abde+acde+bcde
    ==> i[5]=abcde-1
    timer=1;
    ideal j=std(i);
    ==> //used time: 7.5 sec
    size(j);                         // number of elements in the std basis
    ==> 11
    degree(j);
    ==> // codimension = 5
    ==> // dimension   = 0
    ==> // degree      = 70

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