Factorization Bounds and Irreducibility Criteria in Hurwitz Series Rings

Abstract

Let R be a principal ideal domain and let $HR$ denote the Hurwitz series ring over R. We first give a recursive splitting formula for a Hurwitz series whose constant coefficient is a product of two coprime nonunits. This construction leads to irreducibility tests for prime-power constant coefficients, decompositions indexed by the distinct prime divisors of the constant coefficient, and upper and lower bounds for the length of every irreducible factorization. Over a discrete valuation domain, the valuation of any selected coefficient yields a sharper length bound. We then introduce the Hurwitz--Newton polygon and prove a product rule on ranges in which the relevant binomial coefficients are units. Primitive one-edge polygons consequently provide Dumas-type irreducibility criteria. Finally, localization is used to combine independent criteria at several prime elements. Examples over Z, localized polynomial rings, and the Gaussian integers illustrate the results.