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.