On rank in algebraic closure
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913216388399104 |
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| author | Lampert, Amichai Ziegler, Tamar |
| author_facet | Lampert, Amichai Ziegler, Tamar |
| contents | Let $ {\mathbf k} $ be a field and $Q\in {\mathbf k}[x_1, \ldots, x_s]$ a form (homogeneous polynomial) of degree $d>1.$ The ${\mathbf k}$-Schmidt rank $rk_{\mathbf k}(Q)$ of $Q$ is the minimal $r$ such that $Q= \sum_{i=1}^r R_iS_i$ with $R_i, S_i \in {\mathbf k}[x_1, \ldots, x_s]$ forms of degree $<d$. When $ {\mathbf k} $ is algebraically closed, this rank is essentially equivalent to the codimension in $ {\mathbf k}^s $ of the singular locus of the variety defined by $ Q, $ known also as the Birch rank of $ Q. $ When $ {\mathbf k} $ is a number field, a finite field or a function field, we give polynomial bounds for $ rk_{\mathbf k}(Q) $ in terms of $ rk_{\bar {\mathbf k}} (Q) $ where $ \bar {\mathbf k} $ is the algebraic closure of $ {\mathbf k}. $ Prior to this work no such bound (even ineffective) was known for $d>4$. This result has immediate consequences for counting integer points (when $ {\mathbf k} $ is a number field) or prime points (when $ {\mathbf k} = \mathbb Q $) of the variety $ \{Q=0\} $ assuming $ rk_{\mathbf k} (Q) $ is large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_05329 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On rank in algebraic closure Lampert, Amichai Ziegler, Tamar Number Theory Commutative Algebra Algebraic Geometry Let $ {\mathbf k} $ be a field and $Q\in {\mathbf k}[x_1, \ldots, x_s]$ a form (homogeneous polynomial) of degree $d>1.$ The ${\mathbf k}$-Schmidt rank $rk_{\mathbf k}(Q)$ of $Q$ is the minimal $r$ such that $Q= \sum_{i=1}^r R_iS_i$ with $R_i, S_i \in {\mathbf k}[x_1, \ldots, x_s]$ forms of degree $<d$. When $ {\mathbf k} $ is algebraically closed, this rank is essentially equivalent to the codimension in $ {\mathbf k}^s $ of the singular locus of the variety defined by $ Q, $ known also as the Birch rank of $ Q. $ When $ {\mathbf k} $ is a number field, a finite field or a function field, we give polynomial bounds for $ rk_{\mathbf k}(Q) $ in terms of $ rk_{\bar {\mathbf k}} (Q) $ where $ \bar {\mathbf k} $ is the algebraic closure of $ {\mathbf k}. $ Prior to this work no such bound (even ineffective) was known for $d>4$. This result has immediate consequences for counting integer points (when $ {\mathbf k} $ is a number field) or prime points (when $ {\mathbf k} = \mathbb Q $) of the variety $ \{Q=0\} $ assuming $ rk_{\mathbf k} (Q) $ is large. |
| title | On rank in algebraic closure |
| topic | Number Theory Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2205.05329 |