How big polynomial systems can we deal with? How do you know when you don't even have to try?
Recently I tried to solve a problem posed in another MO question and ultimately I got stuck at solving system of four polynomial equations in four variables with rather large polynomials. Their degree was 16 and the number of monomial terms between 400 and 650.
Let $P(i,k,n,m)$ be the class of systems of $k$ polynomial equations in $i$ variables where the polynomials are of degree $n$ or less and have $m$ or less number of terms. And similarly let $S(i,k,n,m)$ be the subclass of polynomial system that are somehow symmetric. (I don't want to be more specific; one can take system invariant or covariant with respect to some group action or systems arising from geometrical problems where the underlying geometrical symmetry is somehow reflected in the system of equations.)
Define feasibility $F_1$ of a problem to mean that it can be solved by a current desktop computer within a week. And define feasibility $F_2$ to mean that the problem can be solved by a current faculty supercomputer within a month.
What is the current state of art of $F_1$ and $F_2$ for $P(k,n,m)$ and $S(k,n,m)$? I'm not looking for precise answer, just a rough idea about the limitations of our technology would be sufficient.