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Viro's patchworking and the signed reduced A-discriminant 维罗的拼接和带符号的减a辨别式
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-05-26 DOI: 10.1016/j.jsc.2025.102462
Weixun Deng , J. Maurice Rojas , Máté L. Telek
{"title":"Viro's patchworking and the signed reduced A-discriminant","authors":"Weixun Deng ,&nbsp;J. Maurice Rojas ,&nbsp;Máté L. Telek","doi":"10.1016/j.jsc.2025.102462","DOIUrl":"10.1016/j.jsc.2025.102462","url":null,"abstract":"<div><div>Computing the isotopy type of a hypersurface, defined as the positive real zero set of a multivariate polynomial, is a challenging problem in real algebraic geometry. We focus on the case where the defining polynomial has combinatorially restricted exponent vectors and fixed coefficient signs, enabling faster computation of the isotopy type. In particular, Viro's patchworking provides a polyhedral complex that has the same isotopy type as the hypersurface, for certain choices of the coefficients. So we present properties of the signed support, focusing mainly on the case of n-variate (n+3)-nomials, that ensure all possible isotopy types can be obtained via patchworking. To prove this, we study the signed reduced A-discriminant and show that it has a simple structure if the signed support satisfies some combinatorial conditions.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102462"},"PeriodicalIF":0.6,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144167100","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
Positivity proofs for linear recurrences through contracted cones 缩锥线性递推的正性证明
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-05-22 DOI: 10.1016/j.jsc.2025.102463
Alaa Ibrahim, Bruno Salvy
{"title":"Positivity proofs for linear recurrences through contracted cones","authors":"Alaa Ibrahim,&nbsp;Bruno Salvy","doi":"10.1016/j.jsc.2025.102463","DOIUrl":"10.1016/j.jsc.2025.102463","url":null,"abstract":"<div><div>Deciding the positivity of a sequence defined by a linear recurrence with polynomial coefficients and initial condition is difficult in general. Even in the case of recurrences with constant coefficients, it is known to be decidable only for order up to 5. We consider a large class of linear recurrences of arbitrary order, with polynomial coefficients, for which an algorithm decides positivity for initial conditions outside of a hyperplane. The underlying algorithm constructs a cone, contracted by the recurrence operator, that allows a proof of positivity by induction. The existence and construction of such cones relies on the extension of the classical Perron-Frobenius theory to matrices leaving a cone invariant.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102463"},"PeriodicalIF":0.6,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144147708","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 implicitizations of multi-graded polynomial maps 多阶多项式映射的计算隐式
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-05-20 DOI: 10.1016/j.jsc.2025.102459
Joseph Cummings , Benjamin Hollering
{"title":"Computing implicitizations of multi-graded polynomial maps","authors":"Joseph Cummings ,&nbsp;Benjamin Hollering","doi":"10.1016/j.jsc.2025.102459","DOIUrl":"10.1016/j.jsc.2025.102459","url":null,"abstract":"<div><div>In this paper, we focus on computing the kernel of a map of polynomial rings. This core problem in symbolic computation is known as implicitization. While Gröbner basis methods can be used to solve this problem, these methods can become infeasible as the number of variables increases. In the case when the polynomial map is multigraded, we consider an alternative approach. We first demonstrate how to quickly compute a matrix of maximal rank for which a polynomial map has a positive multigrading. We then describe how minimal generators in each graded component of the kernel can be computed with linear algebra. We have implemented our techniques in Macaulay2 and show that our implementation can compute many generators of low degree in examples where standard techniques have failed. This includes several examples coming from phylogenetics where even a complete list of quadrics and cubics were unknown. When the multigrading refines total degree, our algorithm is <em>embarassingly parallel</em>. A fully parallelized version of our algorithm is in development in both Macaulay2 and OSCAR.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102459"},"PeriodicalIF":0.6,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134496","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
Extremal decompositions of tropical varieties and relations with rigidity theory 热带品种的极值分解及其与刚性理论的关系
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-05-19 DOI: 10.1016/j.jsc.2025.102461
Farhad Babaee, Sean Dewar, James Maxwell
{"title":"Extremal decompositions of tropical varieties and relations with rigidity theory","authors":"Farhad Babaee,&nbsp;Sean Dewar,&nbsp;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}
引用次数: 0
The convex algebraic geometry of higher-rank numerical ranges 高阶数值范围的凸代数几何
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-05-19 DOI: 10.1016/j.jsc.2025.102457
Jonathan Niño-Cortés, Cynthia Vinzant
{"title":"The convex algebraic geometry of higher-rank numerical ranges","authors":"Jonathan Niño-Cortés,&nbsp;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}
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 : 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":"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}
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 : 2025-05-13 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":"2025-05-13","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}
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 : 2025-04-29 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
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