{"title":"单项式曲线对称性的界限","authors":"Giulio Caviglia, Alessio Moscariello, Alessio Sammartano","doi":"10.1090/proc/16862","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma subset-of-or-equal-to double-struck upper N\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:mo>⊆</mml:mo> <mml:mrow> <mml:mi mathvariant=\"double-struck\">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\Gamma \\subseteq \\mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which depends only on the width of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, that is, the difference between the largest and the smallest generator of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8–28]. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"81 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Bounds for syzygies of monomial curves\",\"authors\":\"Giulio Caviglia, Alessio Moscariello, Alessio Sammartano\",\"doi\":\"10.1090/proc/16862\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Gamma subset-of-or-equal-to double-struck upper N\\\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\\\"normal\\\">Γ</mml:mi> <mml:mo>⊆</mml:mo> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Gamma \\\\subseteq \\\\mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Gamma\\\"> <mml:semantics> <mml:mi mathvariant=\\\"normal\\\">Γ</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which depends only on the width of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Gamma\\\"> <mml:semantics> <mml:mi mathvariant=\\\"normal\\\">Γ</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, that is, the difference between the largest and the smallest generator of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Gamma\\\"> <mml:semantics> <mml:mi mathvariant=\\\"normal\\\">Γ</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8–28]. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":\"81 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-04-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16862\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16862","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let Γ⊆N\Gamma \subseteq \mathbb {N} be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of Γ\Gamma which depends only on the width of Γ\Gamma, that is, the difference between the largest and the smallest generator of Γ\Gamma. In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8–28]. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width.
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