Around the 'Fundamental Theorem of Algebra' (extended version)

TL;DR

This paper extends the Fundamental Theorem of Algebra by presenting new probabilistic analogues for real roots of polynomials, Laurent polynomials, and exponential sums, revealing unexpected properties of zeros in these contexts.

Contribution

It introduces several new versions of the FTA, including for Laurent polynomials, complex reductive groups, and exponential sums, expanding the theorem's scope.

Findings

Most zeros of real Laurent polynomials are real

New FTA versions for polynomials on complex reductive groups

Extended probabilistic analogues for exponential sums

Abstract

The Fundamental Theorem of Algebra (FTA) asserts that every complex polynomial has as many complex roots, counted with multiplicities, as its degree. A probabilistic analogue of this theorem for real roots of real polynomials, sometimes referred to as the Kac theorem, was found between 1938 and 1943 by J. Littlewood, A. Offord, and M. Kac. In this paper, we present several more versions of FTA: Kac type FTA for Laurent polynomials in one and many variables, Kac type FTA for polynomials on complex reductive groups arising in the context of compact group representations (similar to Laurent polynomials arising in torus representation theory), and FTA for exponential sums in one and many variables. In the case of Laurent polynomials, the result, even in the one-dimensional case, is unexpected: most of the zeros of a real Laurent polynomial are real. This text is a supplemented and more detailed version of \cite{arx}.