Given a polynomial system, determine a simple set containing all its solutions

Question

Suppose we have a system of complex polynomials $f = (f_1, \ldots, f_n)$, where each $f_i$ can be viewed as a function $\mathbb{C}^n \to \mathbb{C}$. The solutions of $f$ are the points $x \in \mathbb{C}^n$ such that $f(x) = 0$.

There are results bounding the magnitude of the zeros of univariate polynomials. One result for instance is given here. With this result we know that all zeros of a univariate polynomial $p$ are inside a complex disk or radius $R$ centered at the origin. The value $R$ can be determined using only the coefficients of $p$.

In the title, by simple set I mean a set like the above, which is a disk. What I mean by simple is a set which is determined explicitly by the coefficients of the $f_i$'s and has some simple determining rule. I'm looking for a way to know where the solutions are in advance. Sharp results are welcome.

I know there are some results relating this systems with Newton polyhedra, but I only saw results about counting solutions. If there is a connection between Newton polyhedra and the location of the solutions I would be glad to know. If anyone know some relevant article or book please share it here so I can read it.

Thank you very much.

Answer 1

One large area of research which addresses questions of this type is related to the Lee-Yang theorem, which gives a sufficient condition in terms of coefficients for zeros of a polynomial to lie in the complement of the unit polydisk. Closely related questions are about zeros lying in the complement of $\{ z:\Re z_j>0,\,1\leq j\leq n\}$.

References:

1. E. Lieb and A. Sokal, A General Lee-Yang Theorem for One-Component and Multicomponent Ferromagnets, Commun. Math. Phys. 80, 153-179 (1981).

2. D. Ruelle, Is our mathematics natural? The case of equilibrium statistical mechanics. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 1, 259–268.

3. J. Borcea, P. Brändén, Multivariate Pólya-Schur classification problems in the Weyl algebra, Proc. Lond. Math. Soc. (3) 101 (2010), no. 1, 73–104.

Answer 2

Your set can be infinite and unbounded, so you have to be careful how you state the question. Here are some thoughts in the case $n=2$, which may be possible but difficult to generalize. Let $g$ be the resultant in the variable $x_2$ of the polynomials $f_1,f_2$. If $g$ is identically zero, then $f_1=f_2=0$ contains an algebraic curve and is an infinite set. Otherwise, the zeros of $g$ contain the first coordinates of the common zeros of $f_1,f_2$. The coefficients of $g$ are $O(H^{d_1+d_2})$ where $H$ is a bound for the coefficients of $f_1,f_2$ and $d_1,d_2$ are the degrees in $x_2$ of $f_1,f_2$, respectively. The implied constant depends on $d_1,d_2$, so you can apply the one-dimensional case to get a bound on the zeros of $g$.