Yannakakis' theorem relating the extension complexity of a polytope to the size of a nonnegative factorization of its slack matrix is a seminal result in the study of lifts of convex sets. Inspired by this result and the importance of lifts in the setting of integer programming, we show that a similar result holds for the discrete analog of convex polyhedral cones-affine semigroups. We define the notions of the integer slack matrix and a lift of an affine semigroup. We show that many of the characterizations of the slack matrix in the convex cone setting have analogous results in the affine semigroup setting. We also show how slack matrices of affine semigroups can be used to obtain new results in the study of nonnegative integer rank of nonnegative integer matrices.
{"title":"A Yannakakis-type theorem for lifts of affine semigroups","authors":"João Gouveia, Amy Wiebe","doi":"arxiv-2407.14764","DOIUrl":"https://doi.org/arxiv-2407.14764","url":null,"abstract":"Yannakakis' theorem relating the extension complexity of a polytope to the\u0000size of a nonnegative factorization of its slack matrix is a seminal result in\u0000the study of lifts of convex sets. Inspired by this result and the importance\u0000of lifts in the setting of integer programming, we show that a similar result\u0000holds for the discrete analog of convex polyhedral cones-affine semigroups. We\u0000define the notions of the integer slack matrix and a lift of an affine\u0000semigroup. We show that many of the characterizations of the slack matrix in\u0000the convex cone setting have analogous results in the affine semigroup setting.\u0000We also show how slack matrices of affine semigroups can be used to obtain new\u0000results in the study of nonnegative integer rank of nonnegative integer\u0000matrices.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772161","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}
In this paper, we introduce a model category structure in the category of functors from a filtered poset to cochain complexes in which higher limits of functors that take values in $R$-modules can be computed by means of a fibrant replacement. We explicitly describe a procedure to compute the fibrant replacement and, finally, deduce some vanishing bounds for the higher limits.
{"title":"An homotopical algebra approach to the computation of higher limits","authors":"Guille Carrión Santiago","doi":"arxiv-2407.14205","DOIUrl":"https://doi.org/arxiv-2407.14205","url":null,"abstract":"In this paper, we introduce a model category structure in the category of\u0000functors from a filtered poset to cochain complexes in which higher limits of\u0000functors that take values in $R$-modules can be computed by means of a fibrant\u0000replacement. We explicitly describe a procedure to compute the fibrant\u0000replacement and, finally, deduce some vanishing bounds for the higher limits.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741434","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}
Daniele FaenziIMB, Marcos JardimIMECC, Jean Vall ÈsLMAP
We establish generalizations of Saito's criterion for the freeness of divisors in projective spaces that apply both to sequences of several homogeneous polynomials and to divisors on other complete varieties. As an application, the new criterion is applied to several examples, including sequences whose polynomials depend on disjoint sets of variables, some sequences that are equivariant for the action of a linear group, blow-ups of divisors, and certain sequences of polynomials in positive characteristics.
{"title":"A generalized Saito freeness criterion","authors":"Daniele FaenziIMB, Marcos JardimIMECC, Jean Vall ÈsLMAP","doi":"arxiv-2407.14082","DOIUrl":"https://doi.org/arxiv-2407.14082","url":null,"abstract":"We establish generalizations of Saito's criterion for the freeness of\u0000divisors in projective spaces that apply both to sequences of several\u0000homogeneous polynomials and to divisors on other complete varieties. As an\u0000application, the new criterion is applied to several examples, including\u0000sequences whose polynomials depend on disjoint sets of variables, some\u0000sequences that are equivariant for the action of a linear group, blow-ups of\u0000divisors, and certain sequences of polynomials in positive characteristics.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741432","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 refine and extend a result by Tuitman on the supports of a B'ezout identity satisfied by a finite sequence of sparse Laurent polynomials without common zeroes in the toric variety associated to their supports. When the number of these polynomials is one more than the dimension of the ambient space, we obtain a formula for computing the sparse resultant as the determinant of a Koszul type complex.
{"title":"Sparse Nullstellensatz, resultants and determinants of complexes","authors":"Carlos D'Andrea, Gabriela Jeronimo","doi":"arxiv-2407.13450","DOIUrl":"https://doi.org/arxiv-2407.13450","url":null,"abstract":"We refine and extend a result by Tuitman on the supports of a B'ezout\u0000identity satisfied by a finite sequence of sparse Laurent polynomials without\u0000common zeroes in the toric variety associated to their supports. When the\u0000number of these polynomials is one more than the dimension of the ambient\u0000space, we obtain a formula for computing the sparse resultant as the\u0000determinant of a Koszul type complex.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741438","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}
Matías Bender, Laurent Busé, Carles Checa, Elias Tsigaridas
We study the relation between the multigraded Castelnuovo-Mumford regularity of a multihomogeneous ideal $I$ and the multidegrees of a Gr"obner basis of $I$ with respect to the degree reverse lexicographical monomial order in generic coordinates. For the single graded case, forty years ago, Bayer and Stillman unravelled all aspects of this relation, which in turn the use to complexity estimates for the computation with Gr"obner bases. We build on their work to introduce a bounding region of the multidegrees of minimal generators of multigraded Gr"obner bases for $I$. We also use this region to certify the presence of some minimal generators close to its boundary. Finally, we show that, up to a certain shift, this region is related to the multigraded Castelnuovo-Mumford regularity of $I$.
{"title":"Multigraded Castelnuovo-Mumford regularity and Gröbner bases","authors":"Matías Bender, Laurent Busé, Carles Checa, Elias Tsigaridas","doi":"arxiv-2407.13536","DOIUrl":"https://doi.org/arxiv-2407.13536","url":null,"abstract":"We study the relation between the multigraded Castelnuovo-Mumford regularity\u0000of a multihomogeneous ideal $I$ and the multidegrees of a Gr\"obner basis of\u0000$I$ with respect to the degree reverse lexicographical monomial order in\u0000generic coordinates. For the single graded case, forty years ago, Bayer and\u0000Stillman unravelled all aspects of this relation, which in turn the use to\u0000complexity estimates for the computation with Gr\"obner bases. We build on\u0000their work to introduce a bounding region of the multidegrees of minimal\u0000generators of multigraded Gr\"obner bases for $I$. We also use this region to\u0000certify the presence of some minimal generators close to its boundary. Finally,\u0000we show that, up to a certain shift, this region is related to the multigraded\u0000Castelnuovo-Mumford regularity of $I$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741433","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}
In this article, we consider $mathcal{C}$-semigroups in $mathbb{N}^d$. We start with symmetric and almost symmetric $mathcal{C}$-semigroups and prove that these notions are independent of term orders. We further investigate the conductor and the Ap'ery set of a $mathcal{C}$-semigroup with respect to a minimal extremal ray. Building upon this, we extend the notion of reduced type to $mathcal{C}$-semigroups and study its extremal behavior. For all $d$ and fixed $e geq 2d$, we give a class of $mathcal{C}$-semigroups of embedding dimension $e$ such that both the type and the reduced type do not have any upper bound in terms of the embedding dimension. We further explore irreducible decompositions of a $mathcal{C}$-semigroup and give a lower bound on the irreducible components in an irreducible decomposition. Consequently, we deduce that for each positive integer $k$, there exists a $mathcal{C}$-semigroup $S$ such that the number of irreducible components of $S$ is at least $k$. A $mathcal{C}$-semigroup is known as a generalized numerical semigroup when the rational cone spanned by the semigroup is full. We classify all the symmetric generalized numerical semigroups of embedding dimension $2d+1$. Consequently, when $d>1$, we deduce that a generalized numerical semigroup of embedding dimension $2d+1$ is almost symmetric if and only if it is symmetric.
{"title":"On unboundedness of some invariants of $mathcal{C}$-semigroups","authors":"Om Prakash Bhardwaj, Carmelo Cisto","doi":"arxiv-2407.11584","DOIUrl":"https://doi.org/arxiv-2407.11584","url":null,"abstract":"In this article, we consider $mathcal{C}$-semigroups in $mathbb{N}^d$. We\u0000start with symmetric and almost symmetric $mathcal{C}$-semigroups and prove\u0000that these notions are independent of term orders. We further investigate the\u0000conductor and the Ap'ery set of a $mathcal{C}$-semigroup with respect to a\u0000minimal extremal ray. Building upon this, we extend the notion of reduced type\u0000to $mathcal{C}$-semigroups and study its extremal behavior. For all $d$ and\u0000fixed $e geq 2d$, we give a class of $mathcal{C}$-semigroups of embedding\u0000dimension $e$ such that both the type and the reduced type do not have any\u0000upper bound in terms of the embedding dimension. We further explore irreducible\u0000decompositions of a $mathcal{C}$-semigroup and give a lower bound on the\u0000irreducible components in an irreducible decomposition. Consequently, we deduce\u0000that for each positive integer $k$, there exists a $mathcal{C}$-semigroup $S$\u0000such that the number of irreducible components of $S$ is at least $k$. A\u0000$mathcal{C}$-semigroup is known as a generalized numerical semigroup when the\u0000rational cone spanned by the semigroup is full. We classify all the symmetric\u0000generalized numerical semigroups of embedding dimension $2d+1$. Consequently,\u0000when $d>1$, we deduce that a generalized numerical semigroup of embedding\u0000dimension $2d+1$ is almost symmetric if and only if it is symmetric.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717637","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}
Building on results of Bazzoni-v{S}v{t}ov'{i}v{c}ek, we give a complete classification of the frame of smashing ideals for the derived category of a finite dimensional valuation domain. In particular, we give an explicit construction of an infinite family of commutative rings such that the telescope conjecture fails and which generalise an example of Keller. As a consequence, we deduce that the Krull dimension of the Balmer spectrum and the Krull dimension of the smashing spectrum can differ arbitrarily for rigidly-compactly generated tensor-triangulated categories.
{"title":"Classifying smashing ideals in derived categories of valuation domains","authors":"Scott Balchin, Florian Tecklenburg","doi":"arxiv-2407.11791","DOIUrl":"https://doi.org/arxiv-2407.11791","url":null,"abstract":"Building on results of Bazzoni-v{S}v{t}ov'{i}v{c}ek, we give a complete\u0000classification of the frame of smashing ideals for the derived category of a\u0000finite dimensional valuation domain. In particular, we give an explicit\u0000construction of an infinite family of commutative rings such that the telescope\u0000conjecture fails and which generalise an example of Keller. As a consequence,\u0000we deduce that the Krull dimension of the Balmer spectrum and the Krull\u0000dimension of the smashing spectrum can differ arbitrarily for rigidly-compactly\u0000generated tensor-triangulated categories.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717553","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}
Karim Alexander Adiprasito, Stavros Argyrios Papadakis, Vasiliki Petrotou
Recent works of the authors have demonstrated the usefulness of considering moduli spaces of Artinian reductions of a given ring when studying standard graded rings and their Lefschetz properties. This paper illuminates a key aspect of these works, the behaviour of the canonical module under deformations in this moduli space. We demonstrate that even when there is no natural geometry around, we can give a viewpoint that behaves like it, effectively constructing geometry out of nothing, giving interpretation to intersection numbers without cycles. Moreover, we explore some properties of this normalization.
{"title":"The volume intrinsic to a commutative graded algebra","authors":"Karim Alexander Adiprasito, Stavros Argyrios Papadakis, Vasiliki Petrotou","doi":"arxiv-2407.11916","DOIUrl":"https://doi.org/arxiv-2407.11916","url":null,"abstract":"Recent works of the authors have demonstrated the usefulness of considering\u0000moduli spaces of Artinian reductions of a given ring when studying standard\u0000graded rings and their Lefschetz properties. This paper illuminates a key\u0000aspect of these works, the behaviour of the canonical module under deformations\u0000in this moduli space. We demonstrate that even when there is no natural\u0000geometry around, we can give a viewpoint that behaves like it, effectively\u0000constructing geometry out of nothing, giving interpretation to intersection\u0000numbers without cycles. Moreover, we explore some properties of this\u0000normalization.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717636","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}
The two-dimensional Jacobian Conjecture says that a Keller map $f: (x,y) mapsto (p,q) in k[x,y]^2$ having an invertible Jacobian is an automorphism of $k[x,y]$. We prove that there is no Keller map with $[k(x,y): k(p,q)]$ prime.
{"title":"There are no Keller maps having prime degree field extensions","authors":"Vered Moskowicz","doi":"arxiv-2407.13795","DOIUrl":"https://doi.org/arxiv-2407.13795","url":null,"abstract":"The two-dimensional Jacobian Conjecture says that a Keller map $f: (x,y)\u0000mapsto (p,q) in k[x,y]^2$ having an invertible Jacobian is an automorphism of\u0000$k[x,y]$. We prove that there is no Keller map with $[k(x,y): k(p,q)]$ prime.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741435","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}
Puzzles are a versatile combinatorial tool to interpret the Littlewood-Richardson coefficients for Grassmannians. In this paper, we propose the concept of puzzle ideals whose varieties one-one correspond to the tilings of puzzles and present an algebraic framework to construct the puzzle ideals which works with the Knutson-Tao-Woodward puzzle and its $T$-equivariant and $K$-theoretic variants for Grassmannians. For puzzles for which one side is free, we propose the side-free puzzle ideals whose varieties one-one correspond to the tilings of side-free puzzles, and the elimination ideals of the side-free puzzle ideals contain all the information of the structure constants for Grassmannians with respect to the free side. Besides the underlying algebraic importance of the introduction of these puzzle ideals is the computational feasibility to find all the tilings of the puzzles for Grassmannians by solving the defining polynomial systems, demonstrated with illustrative puzzles via computation of Gr"obner bases.
{"title":"Puzzle Ideals for Grassmannians","authors":"Chenqi Mou, Weifeng Shang","doi":"arxiv-2407.10927","DOIUrl":"https://doi.org/arxiv-2407.10927","url":null,"abstract":"Puzzles are a versatile combinatorial tool to interpret the\u0000Littlewood-Richardson coefficients for Grassmannians. In this paper, we propose\u0000the concept of puzzle ideals whose varieties one-one correspond to the tilings\u0000of puzzles and present an algebraic framework to construct the puzzle ideals\u0000which works with the Knutson-Tao-Woodward puzzle and its $T$-equivariant and\u0000$K$-theoretic variants for Grassmannians. For puzzles for which one side is\u0000free, we propose the side-free puzzle ideals whose varieties one-one correspond\u0000to the tilings of side-free puzzles, and the elimination ideals of the\u0000side-free puzzle ideals contain all the information of the structure constants\u0000for Grassmannians with respect to the free side. Besides the underlying algebraic importance of the introduction of these\u0000puzzle ideals is the computational feasibility to find all the tilings of the\u0000puzzles for Grassmannians by solving the defining polynomial systems,\u0000demonstrated with illustrative puzzles via computation of Gr\"obner bases.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717640","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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