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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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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| author | McCullough, Jason |
| author_facet | McCullough, Jason |
| author_sort | McCullough, Jason |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-6f12132a26b74b47b18a218e737a5552 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-6f12132a26b74b47b18a218e737a55522025-02-07T11:19:53ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-05-01362G325125510.5802/crmath.54410.5802/crmath.544Prime Ideals and Three-generated Ideals with Large RegularityMcCullough, Jason0Iowa State University, Department of Mathematics, Ames, IA, USAAnanyan 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.544/ |
| spellingShingle | McCullough, Jason Prime Ideals and Three-generated Ideals with Large Regularity Comptes Rendus. Mathématique |
| title | Prime Ideals and Three-generated Ideals with Large Regularity |
| title_full | Prime Ideals and Three-generated Ideals with Large Regularity |
| title_fullStr | Prime Ideals and Three-generated Ideals with Large Regularity |
| title_full_unstemmed | Prime Ideals and Three-generated Ideals with Large Regularity |
| title_short | Prime Ideals and Three-generated Ideals with Large Regularity |
| title_sort | prime ideals and three generated ideals with large regularity |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.544/ |
| work_keys_str_mv | AT mcculloughjason primeidealsandthreegeneratedidealswithlargeregularity |