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A syzygial method for equidimensional decomposition 等维分解的协同方法
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-04-29 DOI: 10.1016/j.jsc.2025.102455
Rafael Mohr
{"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}
引用次数: 0
Computing finite and infinite free resolutions with Pommaret-like bases 用类波马雷基底计算有限和无限自由分辨率
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-04-28 DOI: 10.1016/j.jsc.2025.102454
Amir Hashemi , Matthias Orth , Werner M. Seiler
{"title":"Computing finite and infinite free resolutions with Pommaret-like bases","authors":"Amir Hashemi ,&nbsp;Matthias Orth ,&nbsp;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}
引用次数: 0
Semantics of division for polynomial solvers 多项式解的除法语义
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-04-22 DOI: 10.1016/j.jsc.2025.102453
Christopher W. Brown
{"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}
引用次数: 0
Classification of primitive quandles of small order 小阶原始角堆的分类
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-04-22 DOI: 10.1016/j.jsc.2025.102452
Dilpreet Kaur, Pushpendra Singh
{"title":"Classification of primitive quandles of small order","authors":"Dilpreet Kaur,&nbsp;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}
引用次数: 0
Decomposition loci of tensors 张量的分解轨迹
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-04-22 DOI: 10.1016/j.jsc.2025.102451
Alessandra Bernardi , Alessandro Oneto , Pierpaola Santarsiero
{"title":"Decomposition loci of tensors","authors":"Alessandra Bernardi ,&nbsp;Alessandro Oneto ,&nbsp;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}
引用次数: 0
Linear preservers of secant varieties and other varieties of tensors 割线变体和其他变体张量的线性保持器
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-04-17 DOI: 10.1016/j.jsc.2025.102449
Fulvio Gesmundo , Young In Han , Benjamin Lovitz
{"title":"Linear preservers of secant varieties and other varieties of tensors","authors":"Fulvio Gesmundo ,&nbsp;Young In Han ,&nbsp;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}
引用次数: 0
The Chow-Lam form 周林式
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-04-15 DOI: 10.1016/j.jsc.2025.102450
Elizabeth Pratt , Bernd Sturmfels
{"title":"The Chow-Lam form","authors":"Elizabeth Pratt ,&nbsp;Bernd Sturmfels","doi":"10.1016/j.jsc.2025.102450","DOIUrl":"10.1016/j.jsc.2025.102450","url":null,"abstract":"<div><div>The classical Chow form encodes any projective variety by one equation. We here introduce the Chow-Lam form for subvarieties of a Grassmannian. By evaluating the Chow-Lam form at twistor coordinates, we obtain universal projection formulas. These were pioneered by Thomas Lam for positroid varieties in the study of amplituhedra, and we develop his approach further. Universal formulas for branch loci are obtained from Hurwitz-Lam forms. Our focus is on computations and applications in geometry.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102450"},"PeriodicalIF":0.6,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869618","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}
引用次数: 0
Partial semiorthogonal decompositions for quiver moduli 颤模的偏半正交分解
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-04-14 DOI: 10.1016/j.jsc.2025.102448
Gianni Petrella
{"title":"Partial semiorthogonal decompositions for quiver moduli","authors":"Gianni Petrella","doi":"10.1016/j.jsc.2025.102448","DOIUrl":"10.1016/j.jsc.2025.102448","url":null,"abstract":"<div><div>We embed several copies of the derived category of a quiver and certain line bundles in the derived category of an associated moduli space of representations, giving the start of a semiorthogonal decomposition. This mirrors the semiorthogonal decompositions of moduli of vector bundles on curves. Our results are obtained with <span>QuiverTools</span>, an open-source package of tools for quiver representations, their moduli spaces and their geometrical properties.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102448"},"PeriodicalIF":0.6,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143839767","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}
引用次数: 0
On arrangements of quadrics in decomposing the parameter space of 3D digitized rigid motions 论分解三维数字化刚性运动参数空间的四边形排列
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-04-14 DOI: 10.1016/j.jsc.2025.102447
Kacper Pluta , Guillaume Moroz , Yukiko Kenmochi , Pascal Romon
{"title":"On arrangements of quadrics in decomposing the parameter space of 3D digitized rigid motions","authors":"Kacper Pluta ,&nbsp;Guillaume Moroz ,&nbsp;Yukiko Kenmochi ,&nbsp;Pascal Romon","doi":"10.1016/j.jsc.2025.102447","DOIUrl":"10.1016/j.jsc.2025.102447","url":null,"abstract":"<div><div>Computing the arrangement of quadrics in 3D is a fundamental problem in symbolic computation, with challenges arising when handling degenerate cases and asymptotic critical values. State-of-the-art methods typically require a generic change of coordinates to manage these asymptotes, rendering certain problems intractable. A specific instance of this challenge appears in digital geometry, where comparing 3D shapes up to isometry requires applying a 3D rigid motion on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> and mapping the result back to <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, a process typically achieved via a digitization operator. However, such motions do not preserve the topology of digital objects, making the analysis of digitized rigid motions crucial. Our main contribution is the decomposition of the 6D parameter space of digitized rigid motions for image patches of radius up to three. This problem reduces to computing the arrangement of up to 741 quadrics, some of which are degenerate. To address the computational challenges, we introduce and implement a new algorithm for computing arrangements of quadrics in 3D, specifically designed to handle degenerate directions and asymptotic critical values. This approach allows us to overcome the limitations of existing methods, making the problem tractable in the context of digital geometry.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102447"},"PeriodicalIF":0.6,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143855157","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}
引用次数: 0
Geometric interpretations of compatibility for fundamental matrices 基本矩阵相容性的几何解释
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-04-02 DOI: 10.1016/j.jsc.2025.102446
Erin Connelly , Felix Rydell
{"title":"Geometric interpretations of compatibility for fundamental matrices","authors":"Erin Connelly ,&nbsp;Felix Rydell","doi":"10.1016/j.jsc.2025.102446","DOIUrl":"10.1016/j.jsc.2025.102446","url":null,"abstract":"<div><div>In recent work, algebraic computational software was used to provide the exact algebraic conditions under which a six-tuple of fundamental matrices, corresponding to 4 images, is compatible, i.e., there exist 4 cameras such that each pair has the appropriate fundamental matrix. It has been further demonstrated that quadruplewise compatibility is sufficient when the number of cameras greater than 4. We expand on these prior results by proving equivalent geometric conditions for compatibility. We find that compatibility can be characterized via the intersections of epipolar lines in one of the images.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102446"},"PeriodicalIF":0.6,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817118","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}
引用次数: 0
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