@FrançoisBrunault My investigation has two steps: First, experiments, where I necessarily use floating-point precision. In a second step, based on the experiments, I will want to prove some general statements, where I will allow any real-numbers as coefficients (even if these can't be simulated any more). I'm guessing there seems to be a dramatic difference in terms of the capability of the methods you have in mind between these two cases. While I, for now, would be interested mostly in the floating-point case to do the experiments, it would be good to know if there are [...]

Thanks, I wasn't aware of Poonen's book. After doing some reading I realized, given how comprehensive the theory is, that it is absolutely necessary to outline my problem in as much detail as possible. I have written a follow-uip question in that regard (mathoverflow.net/questions/361642/… ) , since giving all that detail it seemed outside the scope of this question.

Thank you very much for all these references, in particular Sturmfels' book seems to contain useful theorems for me.

@tim I think I might have some hope to tackle this - perhaps asking for the state-of-the art was too much. I outlined as best as I could the structure my problem has in a follow-up question (to which I linked in the comment above).

@FrançoisBrunault That sounds actually very interesting to me, thank you for pointing that out. In case you are interested, as I learned through this question that the theory is more involved, I wrote a follow-up question, that details my problem more precisely: mathoverflow.net/questions/361642/…

@tim yes, that is indeed problem in positive dimension. I'm looking for any kind of parametrization of solutions that help my isolate specific variables, as mentioned in my edit.

@RP_ Well, I'm not exclusively interested in the reals, but it does consist in the most important case - see my edit.

@RP_ hm... I don't think it would be that helpful to describe how solving polynomial systems arise in my work, since there are no restrictions on the structure of polynomials systems, so I'm still left with the "how can I solve general polynomials systems over the reals". more specifically, I'm investigating certain classes of neural nets and some metric properties of them, that lead to the questions whether some polynomial systems have solutions.

Or are there any "hidden" things that I have to keep in mind, that they don't mention in the Information for Notices Authors section (such as, e.g., one has to be a member of the AMS to have the article accepts, or to be already well-known research)?