Fast checking that overdetermined polynomial system does not have a solution As a result of some inductive procedure for each $n$ I have a system of about $n^2$ polynomial equations with $n$ variables with integer coefficients, which can be precisely computed. As the system is quite seriously overdetermined I suspect that it has no complex solutions, which is what I need to numerically verify.
I am not an expert in the area, but I have seen somewhere that while in general case the complexity of finding Groebner basis grows very fast, for overdetermined systems there are faster aglorithms not even using Groebner bases that in some cases can even produce polynomial growth of complexity.
Also as I have a particular system, are there some probabilistic algorithms that can give lower complexity as I am not interested in theoretically worst possible cases, but rather have a concrete example.
P.S.Question corrected, thanks to Lev Borisov.
P.P.S. The main thing  which I have in mind is to check that the generated ideal contains $1$ either with Groebner bases or without them. However I do not insist on this method of course.
 A: I know of two methods that work well for problems with low dimensionality; i.e. $n \leq 50$. The first method, due to Lasserre, works by transforming the problem into a series of SDP's. You can find an introduction to that method in [1]. There is a software package, called GloptiPoly, that uses this method.
The second method is to apply the branch-and-bound algorithm. Note that this method requires you to somehow reduce your search space to some bounded subset of $\mathbb{R}^n$. This shouldn't be too hard, since you are concerned with polynomials. Anyway, given a bounded space, you use some range computation method (e.g. Interval Arithmetic (IA), Affine Arithmetic (AA), ...) to bound your polynomial. If the bounds rule out a root, you are done. If not, you subdivide the search space and recurse. The basic logic being simple, sophisticated algorithms of this kind employ a lot of accelerating techniques. Some keywords to look for include: Constraint Propagation, Interval Newton Method, Affine Relaxations. See [2] for a detailed explanation of a method employing several of these accelerating techniques.
I don't have a lot of experience with using the first method, so I'm not sure how it will work for your problem. However, I've solved lots of tightly coupled equations with a method of the second kind, and I think it might serve you well for $n \leq 50$.
[1] Chapter 3: Polynomial Optimization, R. Cominetti, F. Facchinei, J.B. Lasserre, Modern Optimization Modelling Techniques.
[2] A Reliable Affine Relaxation Method for Global Optimization,
J. Ninin, F. Messine, P. Hansen, Technical Report, IRIT and Cahiers du GERAD.
