Journal of Symbolic Computation最新文献

{"title":"Slices of stable polynomials and connections to the Grace-Walsh-Szegő theorem","authors":"Sebastian Debus ,&nbsp;Cordian Riener ,&nbsp;Robin Schabert","doi":"10.1016/j.jsc.2025.102488","DOIUrl":"10.1016/j.jsc.2025.102488","url":null,"abstract":"<div><div>Univariate polynomials are called stable with respect to a domain <em>D</em> if all of their roots lie in <em>D</em>. We study linear slices of the space of stable univariate polynomials with respect to a half-plane. We show that a linear slice always contains a stable polynomial with only a few distinct roots. Subsequently, we apply these results to symmetric polynomials and varieties. We show that for varieties defined by few multiaffine symmetric polynomials, the existence of a point in <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with few distinct coordinates is necessary and sufficient for the intersection with <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> to be non-empty. This is at the same time a generalization of the so-called degree principle to stable polynomials and a result similar to Grace-Walsh-Szegő's coincidence theorem.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"133 ","pages":"Article 102488"},"PeriodicalIF":1.1,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144903702","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":"Simultaneous rational number codes: Decoding beyond half the minimum distance with multiplicities and bad primes","authors":"Matteo Abbondati,&nbsp;Eleonora Guerrini,&nbsp;Romain Lebreton","doi":"10.1016/j.jsc.2025.102481","DOIUrl":"10.1016/j.jsc.2025.102481","url":null,"abstract":"<div><div>In the previous work of <span><span>Abbondati et al. (2024)</span></span>, we extended the decoding analysis of interleaved Chinese remainder codes to simultaneous rational number codes. In this work, we build on <span><span>Abbondati et al. (2024)</span></span> by addressing two important scenarios: multiplicities and the presence of bad primes (divisors of denominators). First, we generalize previous results to multiplicity rational codes by considering modular reductions with respect to prime power moduli. Then, using hybrid analysis techniques, we extend our approach to vectors of fractions that may present bad primes.</div><div>Our contributions include: a decoding algorithm for simultaneous rational number reconstruction with multiplicities, a rigorous analysis of the algorithm's failure probability that generalizes several previous results, an extension to a hybrid model handling situations where not all errors can be assumed random, and a unified approach to handle bad primes within multiplicities. The theoretical results provide a comprehensive probabilistic analysis of reconstruction failure in these more complex scenarios, advancing the state of the art in error correction for rational number codes.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102481"},"PeriodicalIF":1.1,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144828646","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":"Subalgebra and Khovanskii bases equivalence","authors":"Colin Alstad ,&nbsp;Michael Burr ,&nbsp;Oliver Clarke ,&nbsp;Timothy Duff","doi":"10.1016/j.jsc.2025.102480","DOIUrl":"10.1016/j.jsc.2025.102480","url":null,"abstract":"<div><div>We study a partial correspondence between two previously-studied analogues of Gröbner bases in the setting of algebras: namely subalgebra bases for quotients of polynomial rings and Khovanskii bases for valued algebras and domains. Our main motivation is to apply the concrete and computational aspects of subalgebra bases for quotient rings to the abstract theory of Khovanskii bases. Our perspective is that most interesting examples of Khovanskii bases can also be realized as subalgebra bases and vice-versa. As part of this correspondence, we extend the theory of subalgebra bases for quotients of polynomial rings to infinitely generated polynomial algebras and study conditions which make this theory effective. We also provide a computation of Newton-Okounkov bodies from the data of subalgebra bases for quotient rings, which illustrates how interpreting Khovanskii bases as subalgebra bases makes them amenable to existing computer algebra tools.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"133 ","pages":"Article 102480"},"PeriodicalIF":1.1,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886194","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":"2025-08-06","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":"The duality of SONC: Advances in circuit-based certificates","authors":"Janin Heuer,&nbsp;Timo de Wolff","doi":"10.1016/j.jsc.2025.102479","DOIUrl":"10.1016/j.jsc.2025.102479","url":null,"abstract":"<div><div>The cone of sums of nonnegative circuits (SONCs) is a subset of the cone of nonnegative polynomials / exponential sums, which has been studied extensively in recent years. In this article, we construct a subset of the SONC cone which we call the DSONC cone. The DSONC cone is naturally derived from the dual SONC cone; membership can be tested via linear programming. We show that the DSONC cone is a proper, full-dimensional cone, we provide a description of its extreme rays, and collect several properties that parallel those of the SONC cone. Moreover, we show that functions in the DSONC cone cannot have real zeros, which yields that DSONC cone does not intersect the boundary of the SONC cone. Furthermore, we discuss the intersection of the DSONC cone with the SOS and SDSOS cones. Finally, we show that circuit functions in the boundary of the DSONC cone are determined by points of equilibria, which hence are the analogues to singular points in the primal SONC cone, and relate the DSONC cone to tropical geometry.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102479"},"PeriodicalIF":1.1,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144826603","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":"Jacobi stability analysis for systems of ODEs with symbolic computation","authors":"Bo Huang ,&nbsp;Dongming Wang ,&nbsp;Xinyu Wang ,&nbsp;Jing Yang","doi":"10.1016/j.jsc.2025.102478","DOIUrl":"10.1016/j.jsc.2025.102478","url":null,"abstract":"<div><div>The classical theory of Kosambi–Cartan–Chern (KCC) developed in differential geometry provides a powerful method for analyzing the behaviors of dynamical systems. In the KCC theory, the properties of a dynamical system are described in terms of five geometrical invariants, of which the second corresponds to the so-called Jacobi stability of the system. Different from that of the Lyapunov stability that has been studied extensively in the literature, the analysis of the Jacobi stability has been investigated more recently using geometrical concepts and tools. It turns out that the existing work on the Jacobi stability analysis remains theoretical and the problem of algorithmic and symbolic treatment of Jacobi stability analysis has yet to be addressed. In this paper, we initiate our study on the problem for a class of ODE systems of arbitrary dimension and propose two algorithmic schemes using symbolic computation to check whether a nonlinear dynamical system may exhibit Jacobi stability. The first scheme, based on the construction of the complex root structure of a characteristic polynomial and on the method of quantifier elimination, is capable of detecting the existence of the Jacobi stability of the given dynamical system. The second algorithmic scheme exploits the method of semi-algebraic system solving and allows one to determine conditions on the parameters for a given dynamical system to have a prescribed number of Jacobi stable fixed points. Several examples are presented to demonstrate the effectiveness of the proposed algorithmic schemes. The computational results on Jacobi stability of these examples are further verified by numerical simulations.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102478"},"PeriodicalIF":1.1,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144766755","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":"Resultants of skew polynomials over division rings","authors":"Alexis Eduardo Almendras Valdebenito ,&nbsp;Jonathan Armando Briones Donoso ,&nbsp;Andrea Luigi Tironi","doi":"10.1016/j.jsc.2025.102476","DOIUrl":"10.1016/j.jsc.2025.102476","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi></math></span> be a division ring. We generalize some of the main well-known results about the resultant of two univariate polynomials to the more general context of an Ore extension <span><math><mi>F</mi><mo>[</mo><mi>x</mi><mo>;</mo><mi>σ</mi><mo>,</mo><mi>δ</mi><mo>]</mo></math></span>. Moreover, some algorithms and Magma programs are given as a numerical application of the main theoretical results of this paper.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102476"},"PeriodicalIF":1.1,"publicationDate":"2025-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724144","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":"2025-07-23","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":"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":"2025-07-23","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":"Tail reduction free term rewriting systems revisited","authors":"Sándor Vágvölgyi","doi":"10.1016/j.jsc.2025.102474","DOIUrl":"10.1016/j.jsc.2025.102474","url":null,"abstract":"<div><div>First we present various undecidability results on numerous subclasses of tail reduction free term rewriting systems which simply follow from the literature review on term rewriting. Then we show that the following problems are undecidable for linear tail reduction free term rewriting systems: the word problem, the existence of normal forms problem, the common ancestor problem, the joinability problem, the normalizing problem, the termination problem, the convergence problem, the reflexive transitive closure of reduction relation inclusion problem, the reflexive transitive closure of reduction relation equality problem, and the reflexive transitive closure of reduction relation proper inclusion problem. Finally, we show that the following problems are undecidable for right-linear trf TRSs: the inductive problem, the congruence relation inclusion problem, the congruence relation equality problem, and the congruence relation proper inclusion problem. In addition, we show that the restrictions of all the problems to ground terms are also undecidable.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102474"},"PeriodicalIF":0.6,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270009","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}