Factorial and Noetherian Subrings of Power Series Rings

TL;DR

This paper investigates specific subrings of power series rings that exhibit Weierstrass Factorization, leading to unique factorization and Noetherian properties, with applications in p-adic zeta functions and potential for broader generalizations.

Contribution

It introduces a class of intermediate subrings between polynomial and power series rings with proven Weierstrass Factorization, unique factorization, and Noetherian properties.

Findings

Subrings have Weierstrass Factorization.

These subrings are Noetherian.

They enable studying p-adic zeta functions.

Abstract

Let $F$ be a field. We show that certain subrings contained between the polynomial ring $F[X] = F[X_1, ..., X_n]$ and the power series ring $F[X][[Y]] = F[X_1, ..., X_n][[Y]]$ have Weierstrass Factorization, which allows us to deduce both unique factorization and the Noetherian property. These intermediate subrings are obtained from elements of $F[X][[Y]]$ by bounding their total $X$-degree above by a positive real-valued monotonic up function $\lambda$ on their $Y$-degree. These rings arise naturally in studying $p$-adic analytic variation of zeta functions over finite fields. Future research into this area may study more complicated subrings in which $Y = (Y_1, >..., Y_m)$ has more than one variable, and for which there are multiple degree functions, $\lambda_1, ..., \lambda_m$. Another direction of study would be to generalize these results to $k$-affinoid algebras.