Pub Date : 2024-11-26DOI: 10.1016/j.jsc.2024.102404
Emanuele Di Bella, Willem A. de Graaf
We describe computational methods for computing the component group of the stabilizer of a nilpotent element in a complex simple Lie algebra. Our algorithms have been implemented in the language of the computer algebra system GAP4. Occasionally we need Gröbner basis computations; for these we use the systems Magma and Singular. The resulting component groups have been made available in the GAP4 package SLA.
{"title":"Computing component groups of stabilizers of nilpotent orbit representatives","authors":"Emanuele Di Bella, Willem A. de Graaf","doi":"10.1016/j.jsc.2024.102404","DOIUrl":"10.1016/j.jsc.2024.102404","url":null,"abstract":"<div><div>We describe computational methods for computing the component group of the stabilizer of a nilpotent element in a complex simple Lie algebra. Our algorithms have been implemented in the language of the computer algebra system <span>GAP</span>4. Occasionally we need Gröbner basis computations; for these we use the systems <span>Magma</span> and <span>Singular</span>. The resulting component groups have been made available in the <span>GAP</span>4 package <span>SLA</span>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102404"},"PeriodicalIF":0.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142747772","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 : 2024-11-14DOI: 10.1016/j.jsc.2024.102401
Miguel A. Marco-Buzunáriz , Ana Romero
Effective homology techniques allow us to compute homology groups of a wide family of topological spaces. By the Whitehead tower method, this can also be used to compute higher homotopy groups. However, some of these techniques (in particular, the Whitehead tower) rely on the assumption that the starting space is simply connected. For some applications, this problem could be circumvented by replacing the space by its universal cover, which is a simply connected space that shares the higher homotopy groups of the initial space. In this paper, we formalize a simplicial construction for the universal cover, and represent it as a twisted Cartesian product.
As we show with some examples, the universal cover of a space with effective homology does not necessarily have effective homology in general. We show two independent sufficient conditions that can ensure it: one is based on a nilpotency property of the fundamental group, and the other one on discrete vector fields.
Some examples showing our implementation of these constructions in both SageMath and Kenzo are shown, together with an approach to compute the homology of the universal cover when the group is Abelian even in some cases where there is no effective homology, using the twisted homology of the space.
{"title":"Computing the homology of universal covers via effective homology and discrete vector fields","authors":"Miguel A. Marco-Buzunáriz , Ana Romero","doi":"10.1016/j.jsc.2024.102401","DOIUrl":"10.1016/j.jsc.2024.102401","url":null,"abstract":"<div><div>Effective homology techniques allow us to compute homology groups of a wide family of topological spaces. By the Whitehead tower method, this can also be used to compute higher homotopy groups. However, some of these techniques (in particular, the Whitehead tower) rely on the assumption that the starting space is simply connected. For some applications, this problem could be circumvented by replacing the space by its universal cover, which is a simply connected space that shares the higher homotopy groups of the initial space. In this paper, we formalize a simplicial construction for the universal cover, and represent it as a twisted Cartesian product.</div><div>As we show with some examples, the universal cover of a space with effective homology does not necessarily have effective homology in general. We show two independent sufficient conditions that can ensure it: one is based on a nilpotency property of the fundamental group, and the other one on discrete vector fields.</div><div>Some examples showing our implementation of these constructions in both SageMath and Kenzo are shown, together with an approach to compute the homology of the universal cover when the group is Abelian even in some cases where there is no effective homology, using the twisted homology of the space.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102401"},"PeriodicalIF":0.6,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1016/j.jsc.2024.102400
Justin Chen, Marc Härkönen, Anton Leykin
Generalizing the concept of the Macaulay inverse system, we introduce a way to describe localizations of an ideal in a polynomial ring. This leads to an approach to the differential primary decomposition as a description of the affine scheme defined by the ideal.
{"title":"Local dual spaces and primary decomposition","authors":"Justin Chen, Marc Härkönen, Anton Leykin","doi":"10.1016/j.jsc.2024.102400","DOIUrl":"10.1016/j.jsc.2024.102400","url":null,"abstract":"<div><div>Generalizing the concept of the Macaulay inverse system, we introduce a way to describe localizations of an ideal in a polynomial ring. This leads to an approach to the differential primary decomposition as a description of the affine scheme defined by the ideal.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102400"},"PeriodicalIF":0.6,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142651925","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 : 2024-11-08DOI: 10.1016/j.jsc.2024.102399
Renat Gontsov , Irina Goryuchkina
As known, any formal power series solution of an algebraic equation is convergent, as well as that of an analytic one. We study the convergence of formal power series solutions of Mahler functional equations , where is an integer and F is a holomorphic function near . Extending Bézivin's theorem from the polynomial case to the case under consideration we prove that all such solutions are also convergent. The Newton polygonal method for finding them is explained.
{"title":"On the existence and convergence of formal power series solutions of nonlinear Mahler equations","authors":"Renat Gontsov , Irina Goryuchkina","doi":"10.1016/j.jsc.2024.102399","DOIUrl":"10.1016/j.jsc.2024.102399","url":null,"abstract":"<div><div>As known, any formal power series solution <span><math><mi>φ</mi><mo>∈</mo><mi>C</mi><mo>[</mo><mo>[</mo><mi>x</mi><mo>]</mo><mo>]</mo></math></span> of an algebraic equation is convergent, as well as that of an analytic one. We study the convergence of formal power series solutions of Mahler functional equations <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>y</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>ℓ</mi></mrow></msup><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><mi>y</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msup><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, where <span><math><mi>ℓ</mi><mo>⩾</mo><mn>2</mn></math></span> is an integer and <em>F</em> is a holomorphic function near <span><math><mn>0</mn><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msup></math></span>. Extending Bézivin's theorem from the polynomial case to the case under consideration we prove that all such solutions are also convergent. The Newton polygonal method for finding them is explained.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102399"},"PeriodicalIF":0.6,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142651941","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 : 2024-10-30DOI: 10.1016/j.jsc.2024.102398
Luis David García Puente , Elizabeth Gross , Heather A. Harrington , Matthew Johnston , Nicolette Meshkat , Mercedes Pérez Millán , Anne Shiu
Motivated by the question of how biological systems maintain homeostasis in changing environments, Shinar and Feinberg introduced in 2010 the concept of absolute concentration robustness (ACR). A biochemical system exhibits ACR in some species if the steady-state value of that species does not depend on initial conditions. Thus, a system with ACR can maintain a constant level of one species even as the initial condition changes. Despite a great deal of interest in ACR in recent years, the following basic question remains open: How can we determine quickly whether a given biochemical system has ACR? Although various approaches to this problem have been proposed, we show that they are incomplete. Accordingly, we present new methods for deciding ACR, which harness computational algebra. We illustrate our results on several biochemical signaling networks.
{"title":"Absolute concentration robustness: Algebra and geometry","authors":"Luis David García Puente , Elizabeth Gross , Heather A. Harrington , Matthew Johnston , Nicolette Meshkat , Mercedes Pérez Millán , Anne Shiu","doi":"10.1016/j.jsc.2024.102398","DOIUrl":"10.1016/j.jsc.2024.102398","url":null,"abstract":"<div><div>Motivated by the question of how biological systems maintain homeostasis in changing environments, Shinar and Feinberg introduced in 2010 the concept of absolute concentration robustness (ACR). A biochemical system exhibits ACR in some species if the steady-state value of that species does not depend on initial conditions. Thus, a system with ACR can maintain a constant level of one species even as the initial condition changes. Despite a great deal of interest in ACR in recent years, the following basic question remains open: How can we determine quickly whether a given biochemical system has ACR? Although various approaches to this problem have been proposed, we show that they are incomplete. Accordingly, we present new methods for deciding ACR, which harness computational algebra. We illustrate our results on several biochemical signaling networks.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102398"},"PeriodicalIF":0.6,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-30DOI: 10.1016/j.jsc.2024.102397
Gleb Pogudin
When using resultants for elimination, one standard issue is that the resultant vanishes if the variety contains components of dimension larger than the expected dimension. J. Canny proposed an elegant construction, generalized characteristic polynomial, to address this issue by symbolically perturbing the system before the resultant computation. Such perturbed resultant would typically involve artefact components only loosely related to the geometry of the variety of interest. For removing these components, J.M. Rojas proposed to take the greatest common divisor of the results of two different perturbations. In this paper, we investigate this construction, and show that the extra components persistent under taking different perturbations must come either from singularities or from positive-dimensional fibers.
{"title":"Persistent components in Canny's generalized characteristic polynomial","authors":"Gleb Pogudin","doi":"10.1016/j.jsc.2024.102397","DOIUrl":"10.1016/j.jsc.2024.102397","url":null,"abstract":"<div><div>When using resultants for elimination, one standard issue is that the resultant vanishes if the variety contains components of dimension larger than the expected dimension. J. Canny proposed an elegant construction, generalized characteristic polynomial, to address this issue by symbolically perturbing the system before the resultant computation. Such perturbed resultant would typically involve artefact components only loosely related to the geometry of the variety of interest. For removing these components, J.M. Rojas proposed to take the greatest common divisor of the results of two different perturbations. In this paper, we investigate this construction, and show that the extra components persistent under taking different perturbations must come either from singularities or from positive-dimensional fibers.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102397"},"PeriodicalIF":0.6,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578233","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 : 2024-10-11DOI: 10.1016/j.jsc.2024.102395
Jane Ivy Coons , Maize Curiel , Elizabeth Gross
The steady-state degree of a chemical reaction network is the number of complex steady-states for generic rate constants and initial conditions. One way to bound the steady-state degree is through the mixed volume of an associated steady-state system. In this work, we show that for partitionable binomial chemical reaction systems, whose resulting steady-state systems are given by a set of binomials and a set of linear (not necessarily binomial) conservation equations, computing the mixed volume is equivalent to finding the volume of a single mixed cell that is the translate of a parallelotope. Additionally, we give a coloring condition on cycle networks to identify reaction systems with binomial steady-state ideals. We highlight both of these theorems using a class of networks referred to as species-overlapping networks and give a formula for the mixed volume of these networks.
{"title":"Mixed volumes of networks with binomial steady-states","authors":"Jane Ivy Coons , Maize Curiel , Elizabeth Gross","doi":"10.1016/j.jsc.2024.102395","DOIUrl":"10.1016/j.jsc.2024.102395","url":null,"abstract":"<div><div>The steady-state degree of a chemical reaction network is the number of complex steady-states for generic rate constants and initial conditions. One way to bound the steady-state degree is through the mixed volume of an associated steady-state system. In this work, we show that for partitionable binomial chemical reaction systems, whose resulting steady-state systems are given by a set of binomials and a set of linear (not necessarily binomial) conservation equations, computing the mixed volume is equivalent to finding the volume of a single mixed cell that is the translate of a parallelotope. Additionally, we give a coloring condition on cycle networks to identify reaction systems with binomial steady-state ideals. We highlight both of these theorems using a class of networks referred to as species-overlapping networks and give a formula for the mixed volume of these networks.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102395"},"PeriodicalIF":0.6,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532191","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}
We determine the number of complex solutions to a nonlinear eigenvalue problem on the Grassmannian in its Plücker embedding. This is motivated by quantum chemistry, where it represents the truncation to single electrons in coupled cluster theory. We prove the formula for the Grassmannian of lines which was conjectured in earlier work with Fabian Faulstich. This rests on the geometry of the graph of a birational parametrization of the Grassmannian. We present a squarefree Gröbner basis for this graph, and we develop connections to toric degenerations from representation theory.
{"title":"Coupled cluster degree of the Grassmannian","authors":"Viktoriia Borovik , Bernd Sturmfels , Svala Sverrisdóttir","doi":"10.1016/j.jsc.2024.102396","DOIUrl":"10.1016/j.jsc.2024.102396","url":null,"abstract":"<div><div>We determine the number of complex solutions to a nonlinear eigenvalue problem on the Grassmannian in its Plücker embedding. This is motivated by quantum chemistry, where it represents the truncation to single electrons in coupled cluster theory. We prove the formula for the Grassmannian of lines which was conjectured in earlier work with Fabian Faulstich. This rests on the geometry of the graph of a birational parametrization of the Grassmannian. We present a squarefree Gröbner basis for this graph, and we develop connections to toric degenerations from representation theory.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102396"},"PeriodicalIF":0.6,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142437932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-10DOI: 10.1016/j.jsc.2024.102394
Peter Paule , Carsten Schneider
We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.
{"title":"Creative telescoping for hypergeometric double sums","authors":"Peter Paule , Carsten Schneider","doi":"10.1016/j.jsc.2024.102394","DOIUrl":"10.1016/j.jsc.2024.102394","url":null,"abstract":"<div><div>We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102394"},"PeriodicalIF":0.6,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142437933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The equivariant nonnegativity versus sums of squares question has been solved for any infinite series of essential reflection groups but type A. As a first step to a classification, we analyse -invariant quartics. We prove that the cones of invariant sums of squares and nonnegative forms are equal if and only if the number of variables is at most 3 or odd.
对于除 A 型之外的任何无限序列本质反射群，等变非负性与平方和问题都已解决。我们证明，当且仅当变量数至多为 3 或奇数时，不变平方和与非负形式的锥相等。
{"title":"On nonnegative invariant quartics in type A","authors":"Sebastian Debus , Charu Goel , Salma Kuhlmann , Cordian Riener","doi":"10.1016/j.jsc.2024.102393","DOIUrl":"10.1016/j.jsc.2024.102393","url":null,"abstract":"<div><div>The equivariant nonnegativity versus sums of squares question has been solved for any infinite series of essential reflection groups but type <em>A</em>. As a first step to a classification, we analyse <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-invariant quartics. We prove that the cones of invariant sums of squares and nonnegative forms are equal if and only if the number of variables is at most 3 or odd.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102393"},"PeriodicalIF":0.6,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142433151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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