Enclosure of the zero set of polynomials in several complex variables (Q5935452)

The authors give a characterization of a domain in \({{\mathbb C}^{\ast}}^n\) which contains all the zeros of a polynomial \(P\) in \(n\) variables over \({\mathbb C}\). This region, called enclosure of the zero set, is described in terms of Newton polyhedra of \(P\). The authors give many applications, and they discuss various examples. We mention the description of zero-free regions around the \(n\)-torus, a tighter enclosure of the zero set and the computation of the enclosure set for bivariate polynomials. They also study the sign constancy and the positivity of real polynomials over \({\mathbb R}\) and applications to stability. It is proved, for example, that if \[ |a_k-\cdots-a_1|\leq \left(1 - {1 \over {(1+|a_0|)^{1/n}}}\right)^{k_1+k_2+\cdots+k_n}, \] the \(n\)-variate polynomial \[ p(z):= a_0 + \sum_{k\in\mathbb{N}} a_kz^k, \quad a_k\in{\mathbb C}, \;a_0 \neq 0 \] is stable.