C. Galindo, F. Monserrat, C. -J. Moreno-Ávila, J. -J. Moyano-Fernández
We introduce surfaces at infinity, a class of rational surfaces linked to curves with only one place at infinity. The cone of curves of these surfaces is finite polyhedral and minimally generated. We also introduce the $delta$-semigroup of a surface at infinity and consider the set $mathcal{S}$ of surfaces at infinity having the same $delta$-semigroup. We study how the generators of the cone of curves of surfaces in $mathcal{S}$ behave.
{"title":"Surfaces and semigroups at infinity","authors":"C. Galindo, F. Monserrat, C. -J. Moreno-Ávila, J. -J. Moyano-Fernández","doi":"arxiv-2408.15931","DOIUrl":"https://doi.org/arxiv-2408.15931","url":null,"abstract":"We introduce surfaces at infinity, a class of rational surfaces linked to\u0000curves with only one place at infinity. The cone of curves of these surfaces is\u0000finite polyhedral and minimally generated. We also introduce the\u0000$delta$-semigroup of a surface at infinity and consider the set $mathcal{S}$\u0000of surfaces at infinity having the same $delta$-semigroup. We study how the\u0000generators of the cone of curves of surfaces in $mathcal{S}$ behave.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present an effective method for computing parametric primary decomposition via comprehensive Gr"obner systems. In general, it is very difficult to compute a parametric primary decomposition of a given ideal in the polynomial ring with rational coefficients $mathbb{Q}[A,X]$ where $A$ is the set of parameters and $X$ is the set of ordinary variables. One cause of the difficulty is related to the irreducibility of the specialized polynomial. Thus, we introduce a new notion of ``feasibility'' on the stability of the structure of the ideal in terms of its primary decomposition, and we give a new algorithm for computing a so-called comprehensive system consisting of pairs $(C, mathcal{Q})$, where for each parameter value in $C$, the ideal has the stable decomposition $mathcal{Q}$. We may call this comprehensive system a parametric primary decomposition of the ideal. Also, one can also compute a dense set $mathcal{O}$ such that $varphi_alpha(mathcal{Q})$ is a primary decomposition for any $alphain Ccap mathcal{O}$ via irreducible polynomials. In addition, we give several computational examples to examine the effectiveness of our new decomposition.
{"title":"Comprehensive Systems for Primary Decompositions of Parametric Ideals","authors":"Yuki Ishihara, Kazuhiro Yokoyama","doi":"arxiv-2408.15917","DOIUrl":"https://doi.org/arxiv-2408.15917","url":null,"abstract":"We present an effective method for computing parametric primary decomposition\u0000via comprehensive Gr\"obner systems. In general, it is very difficult to\u0000compute a parametric primary decomposition of a given ideal in the polynomial\u0000ring with rational coefficients $mathbb{Q}[A,X]$ where $A$ is the set of\u0000parameters and $X$ is the set of ordinary variables. One cause of the\u0000difficulty is related to the irreducibility of the specialized polynomial.\u0000Thus, we introduce a new notion of ``feasibility'' on the stability of the\u0000structure of the ideal in terms of its primary decomposition, and we give a new\u0000algorithm for computing a so-called comprehensive system consisting of pairs\u0000$(C, mathcal{Q})$, where for each parameter value in $C$, the ideal has the\u0000stable decomposition $mathcal{Q}$. We may call this comprehensive system a\u0000parametric primary decomposition of the ideal. Also, one can also compute a\u0000dense set $mathcal{O}$ such that $varphi_alpha(mathcal{Q})$ is a primary\u0000decomposition for any $alphain Ccap mathcal{O}$ via irreducible\u0000polynomials. In addition, we give several computational examples to examine the\u0000effectiveness of our new decomposition.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $R$ be a ring and $S$ a multiplicative subset of $R$. In this note, we obtain the ACC characterization and Cartan-Eilenberg-Bass theorem for $S$-Noetherian rings. In details, we show that a ring $R$ is an $S$-Noetherian ring if and only if any ascending chain of ideals of $R$ is $S$-stationary, if and only if any direct sum of injective modules is $S$-injective, if and only if any direct limit of injective modules is $S$-injective.
{"title":"A module-theoretic characterization of $S$-Noetherian rings","authors":"Xiaolei Zhang","doi":"arxiv-2408.14781","DOIUrl":"https://doi.org/arxiv-2408.14781","url":null,"abstract":"Let $R$ be a ring and $S$ a multiplicative subset of $R$. In this note, we\u0000obtain the ACC characterization and Cartan-Eilenberg-Bass theorem for\u0000$S$-Noetherian rings. In details, we show that a ring $R$ is an $S$-Noetherian\u0000ring if and only if any ascending chain of ideals of $R$ is $S$-stationary, if\u0000and only if any direct sum of injective modules is $S$-injective, if and only\u0000if any direct limit of injective modules is $S$-injective.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose an effective algorithm that decides if a prime ideal in a polynomial ring over the complex numbers can be transformed into a toric ideal by a linear automorphism of the ambient space. If this is the case, the algorithm computes such a transformation explicitly. The algorithm can compute that all Gaussian graphical models on five vertices that are not initially toric cannot be made toric by any linear change of coordinates. The same holds for all Gaussian conditional independence ideals of undirected graphs on six vertices.
{"title":"Efficiently deciding if an ideal is toric after a linear coordinate change","authors":"Thomas Kahle, Julian Vill","doi":"arxiv-2408.14323","DOIUrl":"https://doi.org/arxiv-2408.14323","url":null,"abstract":"We propose an effective algorithm that decides if a prime ideal in a\u0000polynomial ring over the complex numbers can be transformed into a toric ideal\u0000by a linear automorphism of the ambient space. If this is the case, the\u0000algorithm computes such a transformation explicitly. The algorithm can compute\u0000that all Gaussian graphical models on five vertices that are not initially\u0000toric cannot be made toric by any linear change of coordinates. The same holds\u0000for all Gaussian conditional independence ideals of undirected graphs on six\u0000vertices.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that for a finite set of points $X$ in the projective $n$-space over any field, the Betti number $beta_{n,n+1}$ of the coordinate ring of $X$ is non-zero if and only if $X$ lies on the union of two planes whose sum of dimension is less than $n$. Our proof is direct and short, and the inductive step rests on a combinatorial statement that works over matroids.
{"title":"Betti numbers and linear covers of points","authors":"Hailong Dao, Ben Lund, Sreehari Suresh-Babu","doi":"arxiv-2408.14064","DOIUrl":"https://doi.org/arxiv-2408.14064","url":null,"abstract":"We prove that for a finite set of points $X$ in the projective $n$-space over\u0000any field, the Betti number $beta_{n,n+1}$ of the coordinate ring of $X$ is\u0000non-zero if and only if $X$ lies on the union of two planes whose sum of\u0000dimension is less than $n$. Our proof is direct and short, and the inductive\u0000step rests on a combinatorial statement that works over matroids.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ricardo Burity, Zaqueu Ramos, Aron Simis, Stefan Tohaneanu
Yuzvinsky and Rose-Terao have shown that the homological dimension of the gradient ideal of the defining polynomial of a generic hyperplane arrangement is maximum possible. In this work one provides yet another proof of this result, which in addition is totally different from the one given by Burity-Simis-Tohaneanu. Another main drive of the paper concerns a version of the above result in the case of a product of general forms of arbitrary degrees (in particular, transverse ones). Finally, some relevant cases of non general forms are also contemplated.
{"title":"Rose-Terao-Yuzvinsky theorem for reduced forms","authors":"Ricardo Burity, Zaqueu Ramos, Aron Simis, Stefan Tohaneanu","doi":"arxiv-2408.13579","DOIUrl":"https://doi.org/arxiv-2408.13579","url":null,"abstract":"Yuzvinsky and Rose-Terao have shown that the homological dimension of the\u0000gradient ideal of the defining polynomial of a generic hyperplane arrangement\u0000is maximum possible. In this work one provides yet another proof of this result, which in addition\u0000is totally different from the one given by Burity-Simis-Tohaneanu. Another main\u0000drive of the paper concerns a version of the above result in the case of a\u0000product of general forms of arbitrary degrees (in particular, transverse ones).\u0000Finally, some relevant cases of non general forms are also contemplated.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Doan Trung Cuong, Hailong Dao, David Eisenbud, Toshinori Kobayashi, Claudia Polini, Bernd Ulrich
Let $(R,m,k)$ be a Golod ring. We show a recurrent formula for high syzygies of $k$ interms of previous ones. In the case of embedding dimension at most $2$, we provided complete descriptions of all indecomposable summands of all syzygies of $k$.
{"title":"Syzygies of the residue field over Golod rings","authors":"Doan Trung Cuong, Hailong Dao, David Eisenbud, Toshinori Kobayashi, Claudia Polini, Bernd Ulrich","doi":"arxiv-2408.13425","DOIUrl":"https://doi.org/arxiv-2408.13425","url":null,"abstract":"Let $(R,m,k)$ be a Golod ring. We show a recurrent formula for high syzygies\u0000of $k$ interms of previous ones. In the case of embedding dimension at most\u0000$2$, we provided complete descriptions of all indecomposable summands of all\u0000syzygies of $k$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the families of solvable and nilpotent matroids, and study their realization spaces and their closures. Specifically, we analyze their associated varieties and their irreducible decompositions. Additionally, we study a subfamily of nilpotent matroids, called weak nilpotent matroids, and compute a finite set of defining equations of their associated matroid varieties.
{"title":"Solvable and Nilpotent Matroids: Realizability and Irreducible Decomposition of Their Associated Varieties","authors":"Emiliano Liwski, Fatemeh Mohammadi","doi":"arxiv-2408.12784","DOIUrl":"https://doi.org/arxiv-2408.12784","url":null,"abstract":"We introduce the families of solvable and nilpotent matroids, and study their\u0000realization spaces and their closures. Specifically, we analyze their\u0000associated varieties and their irreducible decompositions. Additionally, we\u0000study a subfamily of nilpotent matroids, called weak nilpotent matroids, and\u0000compute a finite set of defining equations of their associated matroid\u0000varieties.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let R be a commutative ring with identity, and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by $Gamma_I(R)$, is the graph whose vertices are the set ${x in R setminus I | xy in I$ for some $y in R setminus I}$, where distinct vertices x and y are adjacent if and only if $xy in I$. The cozero-divisor graph with respect to I, denoted by $Gamma''_I(R)$, is the graph of $R$ with vertices ${x in R setminus I | xR + I neq R}$, and two distinct vertices x and y are adjacent if and only if $x notin yR + I$ and $y notin xR + I$. In this paper, we introduce and investigate an undirected graph $QGamma''_I(R)$ of R with vertices ${x in R setminus sqrt{I} | xR + I neq R$ and $xR + sqrt{I} = xR + I}$ and two distinct vertices x and y are adjacent if and only if $x notin yR + I$ and $y notin xR + I$.
让 R 是一个具有同一性的交换环，让 I 是 R 的一个理想。R 关于 I 的零因子图，用 $Gamma_I(R)$ 表示，是其顶点为集合 ${x in R setminus I | xy in I$ for some $yin R setminus I}$ 的图，当且仅当 $xy in I$ 时，不同的顶点 x 和 y 是相邻的。与 I 有关的零因子图，用$Gamma''_I(R)$表示，是$R$的图，其顶点为${x in R setminus I | xR+ I neq R}$, 当且仅当 $xnotin yR + I$ 和 $y notin xR + I$ 时，两个不同的顶点 x 和 y 是相邻的。在本文中，我们引入并研究了一个 R 的无向图 $Q（Gamma''_I(R)$，其顶点为 ${x （在 R 中）减去 sqrt{I}| xR + I neq R$ 和 $xR + sqrt{I} = xR + I}$ 并且当且仅当 $x notin yR + I$ 和 $ynotin xR + I$ 时，两个不同的顶点 x 和 y 是相邻的。
{"title":"Ideal-based quasi cozero divisor graph of a commutative ring","authors":"F. Farshadifar","doi":"arxiv-2408.13216","DOIUrl":"https://doi.org/arxiv-2408.13216","url":null,"abstract":"Let R be a commutative ring with identity, and let I be an ideal of R. The\u0000zero-divisor graph of R with respect to I, denoted by $Gamma_I(R)$, is the\u0000graph whose vertices are the set ${x in R setminus I | xy in I$ for some $y\u0000in R setminus I}$, where distinct vertices x and y are adjacent if and only\u0000if $xy in I$. The cozero-divisor graph with respect to I, denoted by\u0000$Gamma''_I(R)$, is the graph of $R$ with vertices ${x in R setminus I | xR\u0000+ I neq R}$, and two distinct vertices x and y are adjacent if and only if $x\u0000notin yR + I$ and $y notin xR + I$. In this paper, we introduce and\u0000investigate an undirected graph $QGamma''_I(R)$ of R with vertices ${x in R\u0000setminus sqrt{I} | xR + I neq R$ and $xR + sqrt{I} = xR + I}$ and two\u0000distinct vertices x and y are adjacent if and only if $x notin yR + I$ and $y\u0000notin xR + I$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jennifer Biermann, Selvi Kara, Augustine O'Keefe, Joseph Skelton, Gabriel Sosa Castillo
In this paper, we investigate the degree of $h$-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of the $h$-polynomial for various fundamental classes of graphs such as paths, cycles, and bipartite graphs. To the best of our knowledge, this marks the first investigation into the combinatorial interpretation of this algebraic invariant. Additionally, we characterize all connected graphs in which the sum of the Castelnuovo-Mumford regularity and the degree of the $h$-polynomial of an edge ideal reaches its maximum value, which is the number of vertices in the graph.
{"title":"Degree of $h$-polynomials of edge ideals","authors":"Jennifer Biermann, Selvi Kara, Augustine O'Keefe, Joseph Skelton, Gabriel Sosa Castillo","doi":"arxiv-2408.12544","DOIUrl":"https://doi.org/arxiv-2408.12544","url":null,"abstract":"In this paper, we investigate the degree of $h$-polynomials of edge ideals of\u0000finite simple graphs. In particular, we provide combinatorial formulas for the\u0000degree of the $h$-polynomial for various fundamental classes of graphs such as\u0000paths, cycles, and bipartite graphs. To the best of our knowledge, this marks\u0000the first investigation into the combinatorial interpretation of this algebraic\u0000invariant. Additionally, we characterize all connected graphs in which the sum\u0000of the Castelnuovo-Mumford regularity and the degree of the $h$-polynomial of\u0000an edge ideal reaches its maximum value, which is the number of vertices in the\u0000graph.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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