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


*

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

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

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