Cyclic n-roots is a popular benchmark problem. It has finitely many solutions iff n is square free. In this case the number of solutions is less than the Bezout bound.
This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.
There were 70 solutions found in 2.737 seconds with 5 variables.
i1 : cyclic(5,QQ) o1 = {a + b + c + d + e, a*b + b*c + c*d + a*e + d*e, a*b*c + b*c*d + a*b*e + ------------------------------------------------------------------------ a*d*e + c*d*e, a*b*c*d + a*b*c*e + a*b*d*e + a*c*d*e + b*c*d*e, ------------------------------------------------------------------------ a*b*c*d*e - 1} o1 : List |
The object cyclic is a method function.