On the Weierstrass Preparation Theorem over General Rings
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Bell, Jason Malcolmson, Peter Okoh, Frank Patel, Yatin |
| author_facet | Bell, Jason Malcolmson, Peter Okoh, Frank Patel, Yatin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10725 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Weierstrass Preparation Theorem over General Rings Bell, Jason Malcolmson, Peter Okoh, Frank Patel, Yatin Commutative Algebra Logic Number Theory 13J05, 13F25, 11J81, 11U05 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. |
| title | On the Weierstrass Preparation Theorem over General Rings |
| topic | Commutative Algebra Logic Number Theory 13J05, 13F25, 11J81, 11U05 |
| url | https://arxiv.org/abs/2504.10725 |