Spectral Factorization of Trigonometric Polynomials and Lattice Geometry

TL;DR

This paper explores spectral factorization of two-variable trigonometric polynomials, proposing a conjecture, proving special cases, and suggesting a method for the general case based on value and root distributions.

Contribution

It formulates a new conjecture on spectral factorization, proves it for specific cases, and introduces an approach to prove it generally using value and root distribution relations.

Findings

Proved the conjecture for special cases

Established relations between value distributions of polynomials

Suggested a new approach for the full conjecture

Abstract

We formulate a conjecture concerning spectral factorization of a class of trigonometric polynomials of two variables and prove it for special cases. Our method uses relations between the distribution of values of a polynomial of two variables and the distributions of values of an associated family of polynomials of one variable. We suggest an approach to prove the full conjecture using relations between the distribution of values and the distribution of roots of polynomials.