Comparing invariants of toric ideals of bipartite graphs

Proceedings of the American Mathematical Society, Series B Pub Date : 2023-03-26 DOI:10.1090/bproc/174

K. Bhaskara, A. Tuyl

{"title":"Comparing invariants of toric ideals of bipartite graphs","authors":"K. Bhaskara, A. Tuyl","doi":"10.1090/bproc/174","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a finite simple graph and let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I Subscript upper G\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>I</mml:mi>\n <mml:mi>G</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">I_G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> denote its associated toric ideal in the polynomial ring <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\n <mml:semantics>\n <mml:mi>R</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. For each integer <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>n</mml:mi>\n <mml:mo>≥</mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">n\\geq 2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, we completely determine all the possible values for the tuple <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis r e g left-parenthesis upper R slash upper I Subscript upper G Baseline right-parenthesis comma degree left-parenthesis h Subscript upper R slash upper I Sub Subscript upper G Subscript Baseline left-parenthesis t right-parenthesis right-parenthesis comma p d i m left-parenthesis upper R slash upper I Subscript upper G Baseline right-parenthesis comma d e p t h left-parenthesis upper R slash upper I Subscript upper G Baseline right-parenthesis comma dimension left-parenthesis upper R slash upper I Subscript upper G Baseline right-parenthesis right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>reg</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>R</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:msub>\n <mml:mi>I</mml:mi>\n <mml:mi>G</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mi>deg</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>h</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>R</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:msub>\n <mml:mi>I</mml:mi>\n <mml:mi>G</mml:mi>\n </mml:msub>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mi>pdim</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>R</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:msub>\n <mml:mi>I</mml:mi>\n <mml:mi>G</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mi>depth</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>R</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:msub>\n <mml:mi>I</mml:mi>\n <mml:mi>G</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mi>dim</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>R</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:msub>\n <mml:mi>I</mml:mi>\n <mml:mi>G</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(\\operatorname {reg}(R/I_G), \\deg (h_{R/I_G}(t)), \\operatorname {pdim}(R/I_G), \\operatorname {depth}(R/I_G), \\dim (R/I_G))</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> when <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a connected bipartite graph on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\n <mml:semantics>\n <mml:mi>n</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> vertices.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"12 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/174","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 4

Abstract

Let G G be a finite simple graph and let I G I_G denote its associated toric ideal in the polynomial ring R R . For each integer n ≥ 2 n\geq 2 , we completely determine all the possible values for the tuple ( reg ⁡ ( R / I G ) , deg ⁡ ( h R / I G ( t ) ) , pdim ⁡ ( R / I G ) , depth ⁡ ( R / I G ) , dim ⁡ ( R / I G ) ) (\operatorname {reg}(R/I_G), \deg (h_{R/I_G}(t)), \operatorname {pdim}(R/I_G), \operatorname {depth}(R/I_G), \dim (R/I_G)) when G G is a connected bipartite graph on n n vertices.