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.

1$\begingroup$ How dense are your polynomials? $\endgroup$ – Igor Rivin Aug 11 '11 at 15:29

$\begingroup$ 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 Rivin Aug 11 '11 at 15:38

$\begingroup$ Take a look at the implementation of TarskiSeidenberg Theorem explained at xorshammer.com/2009/05/14/asuiteofcoollogicprograms I do not know whether this particular implementation is efficient, but it seems worth trying (easy to install, ...). $\endgroup$ – boumol Aug 11 '11 at 16:43

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 Bukh Aug 11 '11 at 18:32

$\begingroup$ If the OP's problem is "generic", then, since the number of equations equals the number of unknowns, the solution is a zerodimensional 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 subresultants. $\endgroup$ – Igor Rivin Aug 11 '11 at 21:47
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.

$\begingroup$ 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. $\endgroup$ – Andrew Aug 11 '11 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 Tal Aug 12 '11 at 19:52