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Geometric complexity theory for product-plus-power 乘积加幂的几何复杂性理论
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2026-01-01 Epub Date: 2025-05-28 DOI: 10.1016/j.jsc.2025.102458
Pranjal Dutta , Fulvio Gesmundo , Christian Ikenmeyer , Gorav Jindal , Vladimir Lysikov
{"title":"Geometric complexity theory for product-plus-power","authors":"Pranjal Dutta ,&nbsp;Fulvio Gesmundo ,&nbsp;Christian Ikenmeyer ,&nbsp;Gorav Jindal ,&nbsp;Vladimir Lysikov","doi":"10.1016/j.jsc.2025.102458","DOIUrl":"10.1016/j.jsc.2025.102458","url":null,"abstract":"<div><div>According to Kumar's recent surprising result (ToCT'20), a small border Waring rank implies that the polynomial can be approximated as a sum of a constant and a small product of linear polynomials. We prove the converse of Kumar's result and establish a tight connection between border Waring rank and the model of computation in Kumar's result. In this way, we obtain a new formulation of border Waring rank, up to a factor of the degree.</div><div>We connect this new formulation to the orbit closure problem of the product-plus-power polynomial. We study this orbit closure from two directions:</div><div>1. We deborder this orbit closure and some related orbit closures, i.e., prove all points in the orbit closure have small non-border algebraic branching programs.</div><div>2. We fully implement the geometric complexity theory approach against the power sum by generalizing the ideas of Ikenmeyer-Kandasamy (STOC'20) to this new orbit closure. In this way, we obtain new multiplicity obstructions that are constructed from just the symmetries of the polynomials.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102458"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144223224","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
Moment varieties of the inverse Gaussian and gamma distributions are nondefective 反高斯分布和伽马分布的矩变化是无缺陷的
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2026-01-01 Epub Date: 2025-05-19 DOI: 10.1016/j.jsc.2025.102460
Oskar Henriksson , Kristian Ranestad , Lisa Seccia , Teresa Yu
{"title":"Moment varieties of the inverse Gaussian and gamma distributions are nondefective","authors":"Oskar Henriksson ,&nbsp;Kristian Ranestad ,&nbsp;Lisa Seccia ,&nbsp;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":"2026-01-01","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}
引用次数: 0
How to generate all possible rational Wilf–Zeilberger forms? 如何生成所有可能的理性Wilf-Zeilberger形式？
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2026-01-01 Epub Date: 2025-06-11 DOI: 10.1016/j.jsc.2025.102473
Shaoshi Chen , Christoph Koutschan , Yisen Wang
{"title":"How to generate all possible rational Wilf–Zeilberger forms?","authors":"Shaoshi Chen ,&nbsp;Christoph Koutschan ,&nbsp;Yisen Wang","doi":"10.1016/j.jsc.2025.102473","DOIUrl":"10.1016/j.jsc.2025.102473","url":null,"abstract":"<div><div>Wilf–Zeilberger pairs are fundamental in the algorithmic theory of Wilf and Zeilberger for computer-generated proofs of combinatorial identities. Wilf–Zeilberger forms are their high-dimensional generalizations, which can be used for proving and discovering convergence acceleration formulas. This paper presents a structural description of all possible rational such forms, which can be viewed as an additive analog of the classical Ore–Sato theorem. Based on this analog, we show a structural decomposition of so-called multivariate hyperarithmetic expressions, which extend multivariate hypergeometric terms to the additive setting.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102473"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322029","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
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 : 2025-11-01 Epub 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":"2025-11-01","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
Quantifier elimination for normal cone computations 用于正常锥体计算的量词消除
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-11-01 Epub Date: 2025-05-13 DOI: 10.1016/j.jsc.2025.102456
Michael Mandlmayr , Ali K. Uncu
{"title":"Quantifier elimination for normal cone computations","authors":"Michael Mandlmayr ,&nbsp;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":"2025-11-01","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}
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 : 2025-11-01 Epub 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":"2025-11-01","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
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 : 2025-11-01 Epub 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":"2025-11-01","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
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 : 2025-11-01 Epub 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":"2025-11-01","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
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 : 2025-11-01 Epub 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":"2025-11-01","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
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 : 2025-11-01 Epub 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":"2025-11-01","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
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