# Quick algorithm for finding real solutions for a system polynomial equations

Would you please recommend a computer program that could give quick answer Yes/No for the question: does there exist a real solution of a given system of polynomial equations with integer coefficients. I will need it to solve a huge list of such systems, each of them is over $\mathbb{R}^6$, has $6$ equations of degree $4$ or less. Coefficients are also quite small. Maple's Triangularize procedure for most of the cases works too long, so applying it for big list is almost impossible.

• How dense are your polynomials? – Igor Rivin Aug 11 '11 at 15:29
• Also, what do the monomials look like (that is, do variables actually appear with degree 4, or is the degree of any given variable bounded by 1 or 2?) – Igor Rivin Aug 11 '11 at 15:38
• Take a look at the implementation of Tarski-Seidenberg Theorem explained at xorshammer.com/2009/05/14/a-suite-of-cool-logic-programs I do not know whether this particular implementation is efficient, but it seems worth trying (easy to install, ...). – boumol Aug 11 '11 at 16:43
• I asked a similar question at mathoverflow.net/questions/1493/…, have a look at the answers. QEPCAD somewhat faster than Maple/Mathematica, but less pleasant to use. If you problem allows, try solving over complex numbers, and then restrict. – Boris Bukh Aug 11 '11 at 18:32
• If the OP's problem is "generic", then, since the number of equations equals the number of unknowns, the solution is a zero-dimensional variety, the number of complex points of which is bounded by the product of the degrees, or 4096, so the problem should not be so bad by sub-resultants. – Igor Rivin Aug 11 '11 at 21:47

• Mathematica and Maple also can find Groebner basis quite effectively. In Mathematica there is also FindInstance[], which finds an instance of solution or gives an empty list if it can't find any. I don't know if it work always. May be a combination of both could help. – Andrew Aug 11 '11 at 18:03