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 poly.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