Hurwitz级数环中的因式分解界与不可约性准则
Factorization Bounds and Irreducibility Criteria in Hurwitz Series Rings
AI总结：
该研究针对Hurwitz级数环，给出递归分解公式、因式分解长度界、Hurwitz-Newton多边形及不可约性准则，结合局部化方法整合多素元准则，并用整数环等实例验证结果。
AI中文摘要：
设R为主理想整环，HR表示R上的Hurwitz级数环。我们首先给出常系数为两个互素非单位乘积的Hurwitz级数的递归分解公式，该构造引出了素幂常系数的不可约性测试、按常系数不同素因子索引的分解，以及每个不可约因式分解长度的上下界；在离散赋值整环上，任意选定系数的赋值能给出更精确的长度界。随后，我们引入Hurwitz-Newton多边形，证明了相关二项式系数为单位的区间上的乘积法则，因此本原单边多边形提供了Dumas型不可约性准则。最后，利用局部化结合多个素元处的独立准则，整数环、局部化多项式环及高斯整数上的例子说明了所得结果。
英文摘要：
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.
