Journal of Symbolic Computation最新文献

{"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":"2026-01-01","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}

{"title":"Computing positive tropical varieties and lower bounds on the number of positive roots","authors":"Kemal Rose ,&nbsp;Máté L. Telek","doi":"10.1016/j.jsc.2025.102477","DOIUrl":"10.1016/j.jsc.2025.102477","url":null,"abstract":"<div><div>We present two effective tools for computing the positive tropicalization of an algebraic variety. First, we outline conditions under which the initial ideal can be used to compute the positive tropicalization, offering a real analogue to the Fundamental Theorem of Tropical Geometry. Additionally, under certain technical assumptions, we provide a real version of the Transverse Intersection Theorem. Building on these results, we propose an algorithm to compute a combinatorial bound on the number of positive real roots of a system of parametrized polynomial equations. Furthermore, we discuss how this combinatorial bound can be applied to study the number of positive steady states of chemical reaction networks.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102477"},"PeriodicalIF":1.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724147","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}

{"title":"ML degrees of Brownian motion tree models: Star trees and root invariance","authors":"Jane Ivy Coons ,&nbsp;Shelby Cox ,&nbsp;Aida Maraj ,&nbsp;Ikenna Nometa","doi":"10.1016/j.jsc.2025.102482","DOIUrl":"10.1016/j.jsc.2025.102482","url":null,"abstract":"<div><div>A Brownian motion tree (BMT) model is a Gaussian model whose associated set of covariance matrices is linearly constrained according to common ancestry in a phylogenetic tree. We study the complexity of inferring the maximum likelihood (ML) estimator for a BMT model by computing its ML-degree. Our main result is that the ML-degree of the BMT model on a star tree with <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span> leaves is <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>3</mn></math></span>, which was previously conjectured by Améndola and Zwiernik. We also prove that the ML-degree of a BMT model is independent of the choice of the root. The proofs rely on the toric geometry of concentration matrices in a BMT model. Toward this end, we produce a combinatorial formula for the determinant of the concentration matrix of a BMT model, which generalizes the Cayley-Prüfer theorem to complete graphs with weights given by a tree.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102482"},"PeriodicalIF":1.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144809587","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}

{"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":"2026-01-01","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}

{"title":"Hypergeometric solutions of linear difference systems","authors":"Moulay Barkatou ,&nbsp;Mark van Hoeij ,&nbsp;Johannes Middeke ,&nbsp;Yi Zhou","doi":"10.1016/j.jsc.2025.102475","DOIUrl":"10.1016/j.jsc.2025.102475","url":null,"abstract":"<div><div>We extend Petkovšek's algorithm for computing hypergeometric solutions of scalar difference equations to the case of difference systems <span><math><mi>τ</mi><mo>(</mo><mi>Y</mi><mo>)</mo><mo>=</mo><mi>M</mi><mi>Y</mi></math></span>, with <span><math><mi>M</mi><mo>∈</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span>, where <em>τ</em> is the shift operator. Hypergeometric solutions are solutions of the form <em>γP</em> where <span><math><mi>P</mi><mo>∈</mo><mi>C</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> and <em>γ</em> is a hypergeometric term over <span><math><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, i.e. <span><math><mi>τ</mi><mo>(</mo><mi>γ</mi><mo>)</mo><mo>/</mo><mi>γ</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. Our contributions concern efficient computation of a set of candidates for <span><math><mi>τ</mi><mo>(</mo><mi>γ</mi><mo>)</mo><mo>/</mo><mi>γ</mi></math></span> which we write as <span><math><mi>λ</mi><mo>=</mo><mi>c</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>B</mi></mrow></mfrac></math></span> with monic <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mi>C</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, <span><math><mi>c</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. Factors of the denominators of <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> and <em>M</em> give candidates for <em>A</em> and <em>B</em>, while another algorithm is needed for <em>c</em>. We use super-reduction algorithm to compute candidates for <em>c</em>, as well as other ingredients to reduce the list of candidates for <span><math><mi>A</mi><mo>/</mo><mi>B</mi></math></span>. To further reduce the number of candidates <span><math><mi>A</mi><mo>/</mo><mi>B</mi></math></span>, we bound the <em>type</em> of <span><math><mi>A</mi><mo>/</mo><mi>B</mi></math></span> by bounding <em>local types</em>. Our algorithm has been implemented in Maple and experiments show that our implementation can handle systems of high dimension, which is useful for factoring operators.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102475"},"PeriodicalIF":1.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144722672","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}

{"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":"2026-01-01","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}

{"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":"2026-01-01","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}

{"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":"2026-01-01","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}

{"title":"Testing containment of tropical hypersurfaces within polynomial complexity","authors":"Dima Grigoriev","doi":"10.1016/j.jsc.2025.102472","DOIUrl":"10.1016/j.jsc.2025.102472","url":null,"abstract":"<div><div>For tropical <em>n</em>-variable polynomials <span><math><mi>f</mi><mo>,</mo><mi>g</mi></math></span> a criterion of containment for tropical hypersurfaces <span><math><mrow><mi>Trop</mi></mrow><mo>(</mo><mi>f</mi><mo>)</mo><mo>⊂</mo><mrow><mi>Trop</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo></math></span> is provided in terms of their Newton polyhedra <span><math><mi>N</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>,</mo><mi>N</mi><mo>(</mo><mi>g</mi><mo>)</mo><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>. Namely, <span><math><mrow><mi>Trop</mi></mrow><mo>(</mo><mi>f</mi><mo>)</mo><mo>⊂</mo><mrow><mi>Trop</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo></math></span> iff for every vertex <em>v</em> of <span><math><mi>N</mi><mo>(</mo><mi>g</mi><mo>)</mo></math></span> there exists a unique vertex <em>w</em> of <span><math><mi>N</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> such that for the tangent cones it holds <span><math><mi>v</mi><mo>−</mo><mi>w</mi><mo>+</mo><mi>N</mi><msub><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mrow><mi>w</mi></mrow></msub><mo>⊆</mo><mi>N</mi><msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mrow><mi>v</mi></mrow></msub></math></span>. Relying on this criterion an algorithm is designed which tests whether <span><math><mrow><mi>Trop</mi></mrow><mo>(</mo><mi>f</mi><mo>)</mo><mo>⊂</mo><mrow><mi>Trop</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo></math></span> within polynomial complexity.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102472"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270130","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}

{"title":"Constructively describing orbit spaces of finite groups by few inequalities","authors":"Philippe Moustrou ,&nbsp;Cordian Riener ,&nbsp;Robin Schabert","doi":"10.1016/j.jsc.2025.102471","DOIUrl":"10.1016/j.jsc.2025.102471","url":null,"abstract":"<div><div>Let <em>G</em> be a finite group acting linearly on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. A celebrated Theorem of Procesi and Schwarz gives an explicit description of the orbit space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>/</mo><mo>/</mo><mi>G</mi></math></span> as a basic closed semi-algebraic set. We give a new proof of this statement and another description as a basic closed semi-algebraic set using elementary tools from real algebraic geometry. Bröcker was able to show that the number of inequalities needed to describe the orbit space generically depends only on the group <em>G</em>. Here, we construct such inequalities explicitly for abelian groups and in the case where only one inequality is needed. Furthermore, we answer an open question raised by Bröcker concerning the genericity of his result.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102471"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144223225","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}