On the Weierstrass Preparation Theorem over General Rings

TL;DR

This work classifies rings for which a Weierstrass-type factorization of power series holds. It shows the strong Weierstrass property holds precisely for finite products of complete local principal ideal rings, while the Weierstrass property for Noetherian rings holds exactly for finite products of complete local Noetherian rings. It then develops a dichotomy for countable rings, proves a broad transcendence criterion for zeros of $p$-adic power series, and establishes undecidability of deciding such factorizations in $R[[x]]$ for infinite finitely generated rings. Together, the results illuminate structural constraints on rings enabling analytic–algebraic factorization and reveal fundamental limits on algorithmic decidability in power-series contexts.

Abstract

We study rings over which an analogue of the Weierstrass preparation theorem holds for power series. We show that a commutative ring $R$ admits a factorization of every power series in $R[[x]]$ as the product of a polynomial and a unit if and only if $R$ is isomorphic to a finite product of complete local principal ideal rings. We also characterize Noetherian rings $R$ for which this factorization holds under the weaker condition that the coefficients of the series generate the unit ideal: this occurs precisely when $R$ is isomorphic to a finite product of complete local Noetherian integral domains. Beyond this, we investigate the failure of Weierstrass-type preparation in finitely generated rings and prove a general transcendence result for zeros of $p$-adic power series, producing a large class of power series over number rings that cannot be written as a polynomial times a unit. Finally, we show that for a finitely generated infinite commutative ring $R$, the decision problem of determining whether an integer power series (with computable coefficients) factors as a polynomial times a unit in $R[[x]]$ is undecidable.

Figures (1)

• Figure 1: A computable bijection $\theta : \mathbb{N} \to \mathbb{Z}^2$ via a spiral walk.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Weierstrass Preparation Theorem
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof