A topological approach to key polynomials
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915760037691392 |
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| author | Nart, Enric Novacoski, Josnei Peruginelli, Giulio |
| author_facet | Nart, Enric Novacoski, Josnei Peruginelli, Giulio |
| contents | In this paper we present characterizations of the sets of key polynomials and abstract key polynomials for a valuation $μ$ of $K(x)$, in terms of (ultrametric) balls in the algebraic closure $\overline K$ of $K$ with respect to $v$, a fixed extension of $μ_{\mid K}$ to $\overline K$. In particular, we show that the ways of augmenting $μ$, in the sense of Mac Lane, are in one-to-one correspondence with the partition of a fixed closed ball $B(a,δ)$ associated to $μ$ into the disjoint union of open balls $B^\circ(a_i,δ)$, modulo the action of the decomposition group of $v$. We also present a similar characterization for the set of limit key polynomials for an increasing family of valuations of $K(x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08357 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A topological approach to key polynomials Nart, Enric Novacoski, Josnei Peruginelli, Giulio Commutative Algebra In this paper we present characterizations of the sets of key polynomials and abstract key polynomials for a valuation $μ$ of $K(x)$, in terms of (ultrametric) balls in the algebraic closure $\overline K$ of $K$ with respect to $v$, a fixed extension of $μ_{\mid K}$ to $\overline K$. In particular, we show that the ways of augmenting $μ$, in the sense of Mac Lane, are in one-to-one correspondence with the partition of a fixed closed ball $B(a,δ)$ associated to $μ$ into the disjoint union of open balls $B^\circ(a_i,δ)$, modulo the action of the decomposition group of $v$. We also present a similar characterization for the set of limit key polynomials for an increasing family of valuations of $K(x)$. |
| title | A topological approach to key polynomials |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2404.08357 |