I'm interested in studying all of the 5-dimensional representations of a certain known Lie algebra. To do this naively, I want to find all of the solutions to a large system of equations, with a few inequations (e.g., a[1,1]<>a[2,2]). I've noticed that calling SolveTools[PolynomialSystem] gives much better performance than solve(...). Since the original system has no equations of degree higher than 2, I suspect that whatever preprocessing that solve(...) is doing is making the system more difficult to solve (higher degree?).
The documentation for solve(...) emphasizes that either all solutions are returned, or it sets _SolutionsMayBeLost. For my purposes this is really important. So:
1) Can SolveTools[PolynomialSystem] similarly guarantee that it has found all solutions? If so, what is the behavior when it cannot find all solutions? (From reading the code, it looks like solve(...) sets _SolutionsMayBeLost if it catches a thrown error "cannot solve"; is this what happens?)
2) Or, is there some way to influence the method that solve(...) uses, so that I can have the documented guarantee of finding all of the solutions?