Intermediate algebras of semialgebraic functions
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916890906984448 |
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| author | Baro, E. Fernando, J. F. Gamboa, J. M. |
| author_facet | Baro, E. Fernando, J. F. Gamboa, J. M. |
| contents | We characterize intermediate $\mathbb{R}$-algebras $A$ between the ring of semialgebraic functions ${\mathcal S}(X)$ and the ring ${\mathcal S}^*(X)$ of bounded semialgebraic functions on a semialgebraic set $X$ as rings of fractions of ${\mathcal S}(X)$. This allows us to compute the Krull dimension of $A$, the transcendence degree over $\mathbb{R}$ of the residue fields of $A$ and to obtain a Łojasiewicz inequality and a Nullstellensatz for archimedean $\mathbb{R}$-algebras $A$. In addition we study intermediate $\mathbb{R}$-algebras generated by proper ideals and we prove an extension theorem for functions in such $\mathbb{R}$-algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13225 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Intermediate algebras of semialgebraic functions Baro, E. Fernando, J. F. Gamboa, J. M. Algebraic Geometry Primary 14P10, 54C30, Secondary 12D15 We characterize intermediate $\mathbb{R}$-algebras $A$ between the ring of semialgebraic functions ${\mathcal S}(X)$ and the ring ${\mathcal S}^*(X)$ of bounded semialgebraic functions on a semialgebraic set $X$ as rings of fractions of ${\mathcal S}(X)$. This allows us to compute the Krull dimension of $A$, the transcendence degree over $\mathbb{R}$ of the residue fields of $A$ and to obtain a Łojasiewicz inequality and a Nullstellensatz for archimedean $\mathbb{R}$-algebras $A$. In addition we study intermediate $\mathbb{R}$-algebras generated by proper ideals and we prove an extension theorem for functions in such $\mathbb{R}$-algebras. |
| title | Intermediate algebras of semialgebraic functions |
| topic | Algebraic Geometry Primary 14P10, 54C30, Secondary 12D15 |
| url | https://arxiv.org/abs/2504.13225 |