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.
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1$\begingroup$ How dense are your polynomials? $\endgroup$– Igor RivinAug 11, 2011 at 15:29
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$\begingroup$ 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?) $\endgroup$– Igor RivinAug 11, 2011 at 15:38
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$\begingroup$ 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, ...). $\endgroup$– boumolAug 11, 2011 at 16:43
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1$\begingroup$ 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. $\endgroup$– Boris BukhAug 11, 2011 at 18:32
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$\begingroup$ 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. $\endgroup$– Igor RivinAug 11, 2011 at 21:47
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Have you thought obtain a Groebner basis of the set of defining polynomials of your system? Maybe, it could simplify considerably the aspect of the system, and you would need a little time to calculate the real solutions of the system. For that, you could use the free software wxMaxima. If you don't have it, you can download in http://maxima.sourceforge.net/download.html.
I would like post this comment just as a comment and not as an answer, but I don't know how to do it.
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$\begingroup$ 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. $\endgroup$– AndrewAug 11, 2011 at 18:03 -
$\begingroup$ I need the process to be very automatical, as there is a huge list I of systems I want to check. Dealing with each of the systems is not a hard problem, but it's a "handmade" work, I won't be able to check all the systems by hand. Most of software can compute Groebner basis, but is there a fast algorithm which can say Yes/No to real roots problem, having a Groebner basis of the ideal? $\endgroup$– Al TalAug 12, 2011 at 19:52