My problem is related to recovering orbits from invariant polynomials, and their ideal of relations.
The invariant polynomials I obtained are:
u = x2 - x y + y2,
v = 2 x6 - 6 x5 y + 15 x4 y2 - 20 x3 y3 + 15 x2 y4 - 6 x y5 + 2 y6 ,
w = x6 - 4 x5 y + 10 x4 y2 - 10 x3 y3 + 5 x2 y4 - 2 x y5 + y6 .
Using the logic from the Cox et al. book I got that the algebraic relation (ideal of relations) between the invariants, which is:
11 u6 - 10 u3 v + 3 (v2 - v w + w2) = 0
Then, using Reduce[] (exact symbolic solver which uses cylindrical algebraic decomposition) from Mathematica I solved for x and y under the assumption x > y > 0, u > 0, v > 0 and w > 0, but I got rather a complex solution.
My question is whether there is a way to try getting something more straightforward than the solution given by Mathematica.
I tried to use Maple's solve function, but it immediately stops without any result of an error message.
My input for Reduce is:
Reduce[{x2 - x y + y2 == u, 2 x6 - 6 x5 y + 15 x4 y2 - 20 x3 y3 + 15 x2 y4 - 6 x y5 + 2 y6 == v, x6 - 4 x5 y + 10 x4 y2 - 10 x3 y3 + 5 x2 y4 - 2 x y5 + y6 == w, 11 u6 - 10 u3 v + 3 (v2 - v w + w2) == 0, x > y > 0, u > 0, v > 0, w > 0}, {x, y}, Complexes]
and my input for solve:
solve({u = x^2 - x*y + y^2, v = 2*x^6 - 6*x^5*y + 15*x^4*y^2 - 20*x^3*y^3 + 15*x^2*y^4 - 6*x*y^5 + 2*y^6, w = x^6 - 4*x^5*y + 10*x^4*y^2 - 10*x^3*y^3 + 5*x^2*y^4 - 2*x*y^5 + y^6, 11*u^6 - 10*u^3*v + 3*v^2 - 3*v*w + 3*w^2 = 0, 0 < u, 0 < v, 0 < w, 0 < x, 0 < y, y < x}, {x, y})
Do you know what I am doing wrong, or what else could I try?