Pub Date : 2025-05-19DOI: 10.1016/j.jsc.2025.102457
Jonathan Niño-Cortés, Cynthia Vinzant
The higher-rank numerical range is a convex compact set generalizing the classical numerical range of a square complex matrix, first appearing in the study of quantum error correction. We will discuss some of the real algebraic and convex geometry of these sets, including a generalization of Kippenhahn's theorem, and describe an algorithm to explicitly calculate the higher-rank numerical range of a given matrix.
{"title":"The convex algebraic geometry of higher-rank numerical ranges","authors":"Jonathan Niño-Cortés, Cynthia Vinzant","doi":"10.1016/j.jsc.2025.102457","DOIUrl":"10.1016/j.jsc.2025.102457","url":null,"abstract":"<div><div>The higher-rank numerical range is a convex compact set generalizing the classical numerical range of a square complex matrix, first appearing in the study of quantum error correction. We will discuss some of the real algebraic and convex geometry of these sets, including a generalization of Kippenhahn's theorem, and describe an algorithm to explicitly calculate the higher-rank numerical range of a given matrix.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102457"},"PeriodicalIF":0.6,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144147707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-19DOI: 10.1016/j.jsc.2025.102461
Farhad Babaee, Sean Dewar, James Maxwell
Extremality and irreducibility constitute fundamental concepts in mathematics, particularly within tropical geometry. While extremal decomposition is typically computationally hard, this article presents a fast algorithm for identifying the extremal decomposition of tropical varieties with rational balanced weightings. Additionally, we explore connections and applications related to rigidity theory. In particular, we prove that a tropical hypersurface is extremal if and only if it has a unique reciprocal diagram up to homothety. We further show that our approach also allows for computing Chow Betti numbers for complete toric varieties.
{"title":"Extremal decompositions of tropical varieties and relations with rigidity theory","authors":"Farhad Babaee, Sean Dewar, James Maxwell","doi":"10.1016/j.jsc.2025.102461","DOIUrl":"10.1016/j.jsc.2025.102461","url":null,"abstract":"<div><div>Extremality and irreducibility constitute fundamental concepts in mathematics, particularly within tropical geometry. While extremal decomposition is typically computationally hard, this article presents a fast algorithm for identifying the extremal decomposition of tropical varieties with rational balanced weightings. Additionally, we explore connections and applications related to rigidity theory. In particular, we prove that a tropical hypersurface is extremal if and only if it has a unique reciprocal diagram up to homothety. We further show that our approach also allows for computing Chow Betti numbers for complete toric varieties.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102461"},"PeriodicalIF":0.6,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144125166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-19DOI: 10.1016/j.jsc.2025.102460
Oskar Henriksson , Kristian Ranestad , Lisa Seccia , Teresa Yu
We show that the parameters of a k-mixture of inverse Gaussian or gamma distributions are algebraically identifiable from the first moments, and rationally identifiable from the first moments. Our proofs are based on Terracini's classification of defective surfaces, careful analysis of the intersection theory of moment varieties, and a recent result on sufficient conditions for rational identifiability of secant varieties by Massarenti–Mella.
{"title":"Moment varieties of the inverse Gaussian and gamma distributions are nondefective","authors":"Oskar Henriksson , Kristian Ranestad , Lisa Seccia , Teresa Yu","doi":"10.1016/j.jsc.2025.102460","DOIUrl":"10.1016/j.jsc.2025.102460","url":null,"abstract":"<div><div>We show that the parameters of a <em>k</em>-mixture of inverse Gaussian or gamma distributions are algebraically identifiable from the first <span><math><mn>3</mn><mi>k</mi><mo>−</mo><mn>1</mn></math></span> moments, and rationally identifiable from the first <span><math><mn>3</mn><mi>k</mi><mo>+</mo><mn>2</mn></math></span> moments. Our proofs are based on Terracini's classification of defective surfaces, careful analysis of the intersection theory of moment varieties, and a recent result on sufficient conditions for rational identifiability of secant varieties by Massarenti–Mella.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102460"},"PeriodicalIF":0.6,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144125175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-13DOI: 10.1016/j.jsc.2025.102456
Michael Mandlmayr , Ali K. Uncu
We present effective procedures to calculate regular normal cones and other related objects using quantifier elimination. This method of normal cone calculations is complementary to computing Lagrangians and it works best at points where the constraint qualifications fail and extra work for other methods becomes inevitable. This method also serves as a tool to calculate the regular co-derivative for semismooth* Newton methods. We list algorithms and their demonstrations of different use cases for this approach.
{"title":"Quantifier elimination for normal cone computations","authors":"Michael Mandlmayr , Ali K. Uncu","doi":"10.1016/j.jsc.2025.102456","DOIUrl":"10.1016/j.jsc.2025.102456","url":null,"abstract":"<div><div>We present effective procedures to calculate regular normal cones and other related objects using quantifier elimination. This method of normal cone calculations is complementary to computing Lagrangians and it works best at points where the constraint qualifications fail and extra work for other methods becomes inevitable. This method also serves as a tool to calculate the regular co-derivative for semismooth* Newton methods. We list algorithms and their demonstrations of different use cases for this approach.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102456"},"PeriodicalIF":0.6,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144090472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-29DOI: 10.1016/j.jsc.2025.102455
Rafael Mohr
Based on a theorem by Vasconcelos, we give an algorithm for equidimensional decomposition of algebraic sets using syzygy computations via Gröbner bases. This algorithm avoids the use of elimination, homological algebra and processing the input equations one-by-one present in previous algorithms. We experimentally demonstrate the practical interest of our algorithm compared to the state of the art.
{"title":"A syzygial method for equidimensional decomposition","authors":"Rafael Mohr","doi":"10.1016/j.jsc.2025.102455","DOIUrl":"10.1016/j.jsc.2025.102455","url":null,"abstract":"<div><div>Based on a theorem by Vasconcelos, we give an algorithm for equidimensional decomposition of algebraic sets using syzygy computations via Gröbner bases. This algorithm avoids the use of elimination, homological algebra and processing the input equations one-by-one present in previous algorithms. We experimentally demonstrate the practical interest of our algorithm compared to the state of the art.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102455"},"PeriodicalIF":0.6,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143903629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-28DOI: 10.1016/j.jsc.2025.102454
Amir Hashemi , Matthias Orth , Werner M. Seiler
Free resolutions are an important tool in algebraic geometry for the structural analysis of modules over polynomial rings and their quotient rings. Minimal free resolutions are unique up to isomorphism and induce homological invariants in the form of Betti numbers. It is known that Pommaret bases of ideals in the polynomial ring induce finite free resolutions and that the Castelnuovo-Mumford regularity and projective dimension can be read off directly from the Pommaret basis. In this article, we generalize this construction to Pommaret-like bases, which are generally smaller. We apply Pommaret-like bases also to infinite resolutions over quotient rings. Over Clements–Lindström rings, we derive bases for the free modules in the resolution using only the Pommaret-like basis. Finally, restricting to monomial ideals in a non-quotient polynomial ring, we derive an explicit formula for the differential of the induced resolution.
{"title":"Computing finite and infinite free resolutions with Pommaret-like bases","authors":"Amir Hashemi , Matthias Orth , Werner M. Seiler","doi":"10.1016/j.jsc.2025.102454","DOIUrl":"10.1016/j.jsc.2025.102454","url":null,"abstract":"<div><div>Free resolutions are an important tool in algebraic geometry for the structural analysis of modules over polynomial rings and their quotient rings. Minimal free resolutions are unique up to isomorphism and induce homological invariants in the form of Betti numbers. It is known that Pommaret bases of ideals in the polynomial ring induce finite free resolutions and that the Castelnuovo-Mumford regularity and projective dimension can be read off directly from the Pommaret basis. In this article, we generalize this construction to Pommaret-like bases, which are generally smaller. We apply Pommaret-like bases also to infinite resolutions over quotient rings. Over Clements–Lindström rings, we derive bases for the free modules in the resolution using only the Pommaret-like basis. Finally, restricting to monomial ideals in a non-quotient polynomial ring, we derive an explicit formula for the differential of the induced resolution.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102454"},"PeriodicalIF":0.6,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143903628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-22DOI: 10.1016/j.jsc.2025.102453
Christopher W. Brown
How to handle division in systems that compute with logical formulas involving what would otherwise be polynomial constraints over the real numbers is a surprisingly difficult question. This paper argues that existing approaches from both the computer algebra and computational logic communities are unsatisfactory for systems that consider the satisfiability of formulas with quantifiers or that perform quantifier elimination. To address this, we propose the notion of the fair-satisfiability of a formula, use it to characterize formulas with divisions that are well-defined, meaning that they adequately guard divisions against division by zero, and provide a translation algorithm that converts a formula with divisions into a purely polynomial formula that is satisfiable if and only if the original formula is fair-satisfiable. This provides a semantics for division with some nice properties, which we describe and prove in the paper.
{"title":"Semantics of division for polynomial solvers","authors":"Christopher W. Brown","doi":"10.1016/j.jsc.2025.102453","DOIUrl":"10.1016/j.jsc.2025.102453","url":null,"abstract":"<div><div>How to handle division in systems that compute with logical formulas involving what would otherwise be polynomial constraints over the real numbers is a surprisingly difficult question. This paper argues that existing approaches from both the computer algebra and computational logic communities are unsatisfactory for systems that consider the satisfiability of formulas with quantifiers or that perform quantifier elimination. To address this, we propose the notion of the <em>fair-satisfiability</em> of a formula, use it to characterize formulas with divisions that are <em>well-defined</em>, meaning that they adequately guard divisions against division by zero, and provide a <em>translation algorithm</em> that converts a formula with divisions into a purely polynomial formula that is satisfiable if and only if the original formula is fair-satisfiable. This provides a semantics for division with some nice properties, which we describe and prove in the paper.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102453"},"PeriodicalIF":0.6,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143900392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-22DOI: 10.1016/j.jsc.2025.102452
Dilpreet Kaur, Pushpendra Singh
In this article, we describe primitive quandles with the help of primitive permutation groups. As a consequence, we enumerate finite non-affine primitive quandles up to order 4096.
{"title":"Classification of primitive quandles of small order","authors":"Dilpreet Kaur, Pushpendra Singh","doi":"10.1016/j.jsc.2025.102452","DOIUrl":"10.1016/j.jsc.2025.102452","url":null,"abstract":"<div><div>In this article, we describe primitive quandles with the help of primitive permutation groups. As a consequence, we enumerate finite non-affine primitive quandles up to order 4096.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102452"},"PeriodicalIF":0.6,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The decomposition locus of a tensor is the set of rank-one tensors appearing in a minimal tensor-rank decomposition of the tensor. For tensors lying on the tangential variety of any Segre variety, but not on the variety itself, we show that the decomposition locus consists of all rank-one tensors except the tangency point only. We also explicitly compute decomposition loci of all tensors belonging to tensor spaces with finitely many orbits with respect to the action of product of general linear groups.
{"title":"Decomposition loci of tensors","authors":"Alessandra Bernardi , Alessandro Oneto , Pierpaola Santarsiero","doi":"10.1016/j.jsc.2025.102451","DOIUrl":"10.1016/j.jsc.2025.102451","url":null,"abstract":"<div><div>The decomposition locus of a tensor is the set of rank-one tensors appearing in a minimal tensor-rank decomposition of the tensor. For tensors lying on the tangential variety of any Segre variety, but not on the variety itself, we show that the decomposition locus consists of all rank-one tensors except the tangency point only. We also explicitly compute decomposition loci of all tensors belonging to tensor spaces with finitely many orbits with respect to the action of product of general linear groups.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102451"},"PeriodicalIF":0.6,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143878859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-17DOI: 10.1016/j.jsc.2025.102449
Fulvio Gesmundo , Young In Han , Benjamin Lovitz
We study the problem of characterizing linear preserver subgroups of algebraic varieties, with a particular emphasis on secant varieties and other varieties of tensors. We introduce a number of techniques built on different geometric properties of the varieties of interest. Our main result is a simple characterization of the linear preservers of secant varieties of Segre varieties in many cases, including for all . We also characterize the linear preservers of several other sets of tensors, including subspace varieties, the variety of slice rank one tensors, symmetric tensors of bounded Waring rank, the variety of biseparable tensors, and hyperdeterminantal surfaces. Computational techniques and applications in quantum information theory are discussed. We provide geometric proofs for several previously known results on linear preservers.
{"title":"Linear preservers of secant varieties and other varieties of tensors","authors":"Fulvio Gesmundo , Young In Han , Benjamin Lovitz","doi":"10.1016/j.jsc.2025.102449","DOIUrl":"10.1016/j.jsc.2025.102449","url":null,"abstract":"<div><div>We study the problem of characterizing linear preserver subgroups of algebraic varieties, with a particular emphasis on secant varieties and other varieties of tensors. We introduce a number of techniques built on different geometric properties of the varieties of interest. Our main result is a simple characterization of the linear preservers of secant varieties of Segre varieties in many cases, including <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><msup><mrow><mo>(</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>×</mo><mi>k</mi></mrow></msup><mo>)</mo></math></span> for all <span><math><mi>r</mi><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>⌊</mo><mi>k</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></msup></math></span>. We also characterize the linear preservers of several other sets of tensors, including subspace varieties, the variety of slice rank one tensors, symmetric tensors of bounded Waring rank, the variety of biseparable tensors, and hyperdeterminantal surfaces. Computational techniques and applications in quantum information theory are discussed. We provide geometric proofs for several previously known results on linear preservers.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102449"},"PeriodicalIF":0.6,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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