Spectral Factorization of Trigonometric Polynomials and Lattice Geometry

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.Comment: This paper has been accepted for publication in Acta Arithmetica, it is a thoroughly revised and retitled version of the arXiv:1110.5277 paper submitted on 24 Oct 2011 that was titled "Spectral Factorization and Lattice Geometry