An Analysis upon Various Applications of Zeros of Complex Univariate and Multivariate Polynomials: A Review: Exploring the Power and Versatility of Zeros of Polynomials

Abstract

Problems in many different areas of mathematics reduce to questions about the zeros of complex univariate and multivariate polynomials. Recently, several significant and seemingly unrelated results relevant to theoretical computer science have benefited from taking this route: they rely on showing, at some level, that a certain univariate or multivariate polynomial has no zeros in a region. This is achieved by inductively constructing the relevant polynomial via a sequence of operations which preserve the property of not having roots in the required region. The goal of this article is to present this viewpoint and to convey why the study of zeros is a natural, powerful, and versatile tool. It is meant to be a gentle introduction for the essential techniques underlying these results, is largely self-contained and aimed at a broad theory audience.