Pub Date : 2025-07-23DOI: 10.1016/j.jsc.2025.102477
Kemal Rose , Máté L. Telek
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.
{"title":"Computing positive tropical varieties and lower bounds on the number of positive roots","authors":"Kemal Rose , 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}
Pub Date : 2025-07-23DOI: 10.1016/j.jsc.2025.102475
Moulay Barkatou , Mark van Hoeij , Johannes Middeke , Yi Zhou
We extend Petkovšek's algorithm for computing hypergeometric solutions of scalar difference equations to the case of difference systems , with , where τ is the shift operator. Hypergeometric solutions are solutions of the form γP where and γ is a hypergeometric term over , i.e. . Our contributions concern efficient computation of a set of candidates for which we write as with monic , . Factors of the denominators of and M give candidates for A and B, while another algorithm is needed for c. We use super-reduction algorithm to compute candidates for c, as well as other ingredients to reduce the list of candidates for . To further reduce the number of candidates , we bound the type of by bounding local types. 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.
{"title":"Hypergeometric solutions of linear difference systems","authors":"Moulay Barkatou , Mark van Hoeij , Johannes Middeke , 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}
Pub Date : 2025-06-11DOI: 10.1016/j.jsc.2025.102474
Sándor Vágvölgyi
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.
{"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}
Pub Date : 2025-06-11DOI: 10.1016/j.jsc.2025.102472
Dima Grigoriev
For tropical n-variable polynomials a criterion of containment for tropical hypersurfaces is provided in terms of their Newton polyhedra . Namely, iff for every vertex v of there exists a unique vertex w of such that for the tangent cones it holds . Relying on this criterion an algorithm is designed which tests whether within polynomial complexity.
{"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":"2025-06-11","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}
Pub Date : 2025-06-11DOI: 10.1016/j.jsc.2025.102473
Shaoshi Chen , Christoph Koutschan , Yisen Wang
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.
{"title":"How to generate all possible rational Wilf–Zeilberger forms?","authors":"Shaoshi Chen , Christoph Koutschan , 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":"2025-06-11","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}
Pub Date : 2025-05-30DOI: 10.1016/j.jsc.2025.102471
Philippe Moustrou , Cordian Riener , Robin Schabert
Let G be a finite group acting linearly on . A celebrated Theorem of Procesi and Schwarz gives an explicit description of the orbit space 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 G. 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.
{"title":"Constructively describing orbit spaces of finite groups by few inequalities","authors":"Philippe Moustrou , Cordian Riener , 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":"2025-05-30","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}
Pub Date : 2025-05-28DOI: 10.1016/j.jsc.2025.102458
Pranjal Dutta , Fulvio Gesmundo , Christian Ikenmeyer , Gorav Jindal , Vladimir Lysikov
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.
We connect this new formulation to the orbit closure problem of the product-plus-power polynomial. We study this orbit closure from two directions:
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.
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.
{"title":"Geometric complexity theory for product-plus-power","authors":"Pranjal Dutta , Fulvio Gesmundo , Christian Ikenmeyer , Gorav Jindal , 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":"2025-05-28","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}
Pub Date : 2025-05-26DOI: 10.1016/j.jsc.2025.102462
Weixun Deng , J. Maurice Rojas , Máté L. Telek
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.
{"title":"Viro's patchworking and the signed reduced A-discriminant","authors":"Weixun Deng , J. Maurice Rojas , 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}
Pub Date : 2025-05-22DOI: 10.1016/j.jsc.2025.102463
Alaa Ibrahim, Bruno Salvy
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.
{"title":"Positivity proofs for linear recurrences through contracted cones","authors":"Alaa Ibrahim, 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}
Pub Date : 2025-05-20DOI: 10.1016/j.jsc.2025.102459
Joseph Cummings , Benjamin Hollering
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 embarassingly parallel. A fully parallelized version of our algorithm is in development in both Macaulay2 and OSCAR.
{"title":"Computing implicitizations of multi-graded polynomial maps","authors":"Joseph Cummings , 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}
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