Logarithmic Newton polygons and polytopes, and the factorization of Dirichlet polynomials

TL;DR

This paper develops new methods to analyze the factorization of Dirichlet polynomials using logarithmic Newton polygons and polytopes, providing irreducibility criteria and bounds without relying on prime factorizations.

Contribution

It introduces novel irreducibility criteria and factorization bounds for Dirichlet polynomials using logarithmic Newton polygons and polytopes, extending classical polynomial results.

Findings

Irreducibility criteria analogous to classical polynomial results.

Upper bounds for multiplicities of irreducible factors.

Excluding intervals for relative degrees of factors.

Abstract

To study a Dirichlet polynomial $f(s)=\frac{a_{m}}{m^{s}}+\cdots +\frac{a_{n}}{n^{s}}$ by regarding it as a multivariate polynomial via the canonical map $\phi$ sending $p_i^{-s}$ to an indeterminate $X_i$, with $p_i$ the $i$th prime number, requires knowing the prime factorizations of all the integers in the support of $f$. We devise several methods to study the factorization of Dirichlet polynomials over unique factorization domains that circumvent the use of $\phi$, and obtain irreducibility criteria that are analogous to the classical results of Sch\"onemann, Eisenstein, Dumas, St\"ackel, Ore and Weisner for polynomials, and to more recent results of Filaseta and Cavachi. Some of the proofs rely on logarithmic versions of the classical Newton polygons. Criteria that use two or more $p$-adic valuations by combining information from different logarithmic Newton polygons of $f$, as well as irreducibility conditions for Dirichlet polynomials that assume a prime or a prime power value are also obtained. We also find excluding intervals for the relative degrees of the factors of a Dirichlet polynomial, and upper bounds for the multiplicities of the irreducible factors, in particular square-free criteria, that use no derivatives. Criteria of absolute irreducibility analogous to results of Ostrowski, Gao and Stepanov-Schmidt are finally provided in the multivariate case by using logarithmic Newton polytopes and logarithmic upper Newton polygons.