Veit> Most of you know lots more about computer algebra than I do. I'm
interested in finding ALL solutions over the complex numbers of the following five equations:
{1 + 1/b + c + b/d + e/c + d/f + f/e == 0, 1 + b + 1/c + d/b + c/e + e/f + f/d == 0, 1 + c/b + 1/d + d/e + e + b/f + f/c == 0, 1 + b/c + d + 1/e + e/d + c/f + f/b == 0, 1 + c/d + d/c + b/e + e/b + 1/f + f == 0}
Here's what I already know:
There exist solutions where all the variables are 6th roots of 1.
There exist solutions where all the variables are 7th roots of 1.
Numerically I can find solutions different from either of the above to very many digits.
When I use Reduce or GroebnerBasis in Mma7 I quickly run out of memory. Any suggestions?
Veit
You can get a lot of slutions just by treewalking with Factor[Resultant[...]]. Down the e=d=1 branch, {{f -> 1, b -> 1, c -> 1/2 (-5 - Sqrt[21])}, {f -> 1, b -> 1, c -> 1/2 (-5 + Sqrt[21])}, {f -> 1, b -> 1/2 (-5 - Sqrt[21]), c -> 1}, {f -> 1, b -> 1/2 (-5 + Sqrt[21]), c -> 1}, {f -> 1/2 (-5 - Sqrt[21]), c -> 1, b -> 1}, {f -> 1/2 (-5 + Sqrt[21]), c -> 1, b -> 1}, {f -> 1/10 (17 - 3 Sqrt[21]), c -> 1/10 (-11 - Sqrt[21]), b -> 1/10 (-11 - Sqrt[21])}, {f -> 1/10 (17 - 3 Sqrt[21]), c -> 1/10 (-11 + Sqrt[21]), b -> 1/10 (-11 + Sqrt[21])}, {f -> 1/10 (17 + 3 Sqrt[21]), c -> 1/10 (-11 - Sqrt[21]), b -> 1/10 (-11 - Sqrt[21])}, {f -> 1/10 (17 + 3 Sqrt[21]), c -> 1/10 (-11 + Sqrt[21]), b -> 1/10 (-11 + Sqrt[21])}, {f -> 1/10 (-11 - Sqrt[21]), c -> 1/10 (17 - 3 Sqrt[21]), b -> 1/10 (-11 - Sqrt[21])}, {f -> 1/10 (-11 - Sqrt[21]), c -> 1/10 (-11 - Sqrt[21]), b -> 1/10 (17 - 3 Sqrt[21])}, {f -> 1/10 (-11 - Sqrt[21]), c -> 1/10 (-11 + Sqrt[21]), b -> 1/10 (17 + 3 Sqrt[21])}, {f -> 1/10 (-11 - Sqrt[21]), c -> 1/10 (17 + 3 Sqrt[21]), b -> 1/10 (-11 + Sqrt[21])}, {f -> 1/10 (-11 + Sqrt[21]), c -> 1/10 (17 - 3 Sqrt[21]), b -> 1/10 (-11 - Sqrt[21])}, {f -> 1/10 (-11 + Sqrt[21]), c -> 1/10 (-11 - Sqrt[21]), b -> 1/10 (17 - 3 Sqrt[21])}, {f -> 1/10 (-11 + Sqrt[21]), c -> 1/10 (-11 + Sqrt[21]), b -> 1/10 (17 + 3 Sqrt[21])}, {f -> 1/10 (-11 + Sqrt[21]), c -> 1/10 (17 + 3 Sqrt[21]), b -> 1/10 (-11 + Sqrt[21])}} plus two more apparently intractable sets with f = irreducible sextics. --rwg