Prime Ideals and Three-generated Ideals with Large Regularity
Ananyan and Hochster proved the existence of a function $\Phi (m,d)$ such that any graded ideal $I$ generated by $m$ forms of degree at most $d$ in a standard graded polynomial ring satisfies $\mathrm{reg}(I) \le \Phi (m,d)$. Relatedly, Caviglia et. al. proved the existence of a function $\Psi (e)$...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.544/ |
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| Summary: | Ananyan and Hochster proved the existence of a function $\Phi (m,d)$ such that any graded ideal $I$ generated by $m$ forms of degree at most $d$ in a standard graded polynomial ring satisfies $\mathrm{reg}(I) \le \Phi (m,d)$. Relatedly, Caviglia et. al. proved the existence of a function $\Psi (e)$ such that any nondegenerate prime ideal $P$ of degree $e$ in a standard graded polynomial ring over an algebraically closed field satisfies $\mathrm{reg}(P) \le \Psi (\deg (P))$. We provide a construction showing that both $\Phi (3,d)$ and $\Psi (e)$ must be at least doubly exponential in $d$ and $e$, respectively. Previously known lower bounds were merely super-polynomial in both cases. |
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| ISSN: | 1778-3569 |