Given finitely many multivariate polynomials with algebraic coefficients that generate a zero-dimensional ideal, is there an easy way to find a nonzero single-variable polynomial in this ideal?
If we compute a Groebner basis in the lexicographic ordering, then the first polynomial in the resulting triangular system will do the trick, but I want an approach that takes less time.
My intuition is that a Groebner basis is not necessary here. For example, suppose the variety is empty. Then any nonzero single-variable polynomial will work for my purposes. Meanwhile, a Groebner basis calculation will spin its wheels to prove that the variety is empty. This is overkill!
Bonus points for a faster-than-Groebner solution that's already implemented in a standard computer algebra system.
