A quantitative Hilbert's basis theorem and the constructive Krull dimension
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915470922219520 |
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| author | Kuroki, Ryota |
| author_facet | Kuroki, Ryota |
| contents | In classical mathematics, Gulliksen has introduced the length of Noetherian modules, and Brookfield has determined the length of Noetherian polynomial rings. Brookfield's result can be regarded as a quantitative version of Hilbert's basis theorem. In this paper, based on the inductive definition of Noetherian modules in constructive algebra, we introduce a constructive version of the length called $α$-Noetherian modules, and present a constructive proof of some results by Brookfield. As a consequence, we obtain a new constructive proof of $\dim K[X_0,\ldots,X_{n-1}]<1+n$ and $\dim\mathbb{Z}[X_0,\ldots,X_{n-1}]<2+n$, where $K$ is a discrete field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00363 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A quantitative Hilbert's basis theorem and the constructive Krull dimension Kuroki, Ryota Rings and Algebras 16P40 (Primary) 03F65 (Secondary) In classical mathematics, Gulliksen has introduced the length of Noetherian modules, and Brookfield has determined the length of Noetherian polynomial rings. Brookfield's result can be regarded as a quantitative version of Hilbert's basis theorem. In this paper, based on the inductive definition of Noetherian modules in constructive algebra, we introduce a constructive version of the length called $α$-Noetherian modules, and present a constructive proof of some results by Brookfield. As a consequence, we obtain a new constructive proof of $\dim K[X_0,\ldots,X_{n-1}]<1+n$ and $\dim\mathbb{Z}[X_0,\ldots,X_{n-1}]<2+n$, where $K$ is a discrete field. |
| title | A quantitative Hilbert's basis theorem and the constructive Krull dimension |
| topic | Rings and Algebras 16P40 (Primary) 03F65 (Secondary) |
| url | https://arxiv.org/abs/2509.00363 |