Journal of Symbolic Computation最新文献

{"title":"Minimal generating sets of large powers of bivariate monomial ideals","authors":"Jutta Rath,&nbsp;Roswitha Rissner","doi":"10.1016/j.jsc.2026.102554","DOIUrl":"10.1016/j.jsc.2026.102554","url":null,"abstract":"<div><div>It is known that for a monomial ideal <em>I</em>, the number of minimal generators, <span><math><mi>μ</mi><mo>(</mo><msup><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>, eventually follows a polynomial pattern for increasing <em>n</em>. In general, little is known about the power at which this pattern emerges. Even less is known about the exact form of the minimal generators after this power. Let <span><math><mi>s</mi><mo>≥</mo><mi>μ</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>(</mo><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></math></span>, where <em>d</em> is a constant bounded above by the maximal <em>x</em>- or <em>y</em>-degree appearing in the set <span><math><mi>G</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> of minimal generators of <em>I</em>. We show that every higher power <span><math><msup><mrow><mi>I</mi></mrow><mrow><mi>s</mi><mo>+</mo><mi>ℓ</mi></mrow></msup></math></span> for any <span><math><mi>ℓ</mi><mo>≥</mo><mn>0</mn></math></span> can be constructed from certain subideals of <span><math><msup><mrow><mi>I</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span>. This provides an explicit description of <span><math><mi>G</mi><mo>(</mo><msup><mrow><mi>I</mi></mrow><mrow><mi>s</mi><mo>+</mo><mi>ℓ</mi></mrow></msup><mo>)</mo></math></span> in terms of <span><math><mi>G</mi><mo>(</mo><msup><mrow><mi>I</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>)</mo></math></span>. Given <span><math><mi>G</mi><mo>(</mo><msup><mrow><mi>I</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>)</mo></math></span>, this construction significantly reduces computational complexity in determining larger powers of <em>I</em>. This further enables us to explicitly compute <span><math><mi>μ</mi><mo>(</mo><msup><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mi>s</mi></math></span> in terms of a linear polynomial in <em>n</em>. We include runtime measurements for the attached implementation in SageMath.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"136 ","pages":"Article 102554"},"PeriodicalIF":1.1,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145996181","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":"Differential operators on homogeneous plane curve singularities","authors":"Julien Sebag","doi":"10.1016/j.jsc.2026.102556","DOIUrl":"10.1016/j.jsc.2026.102556","url":null,"abstract":"<div><div>Let <em>k</em> be a field with characteristic zero. Let <span><math><mi>f</mi><mo>∈</mo><mi>k</mi><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></math></span> be a reduced homogeneous polynomial with degree <span><math><mi>d</mi><mo>≥</mo><mn>1</mn></math></span>. We set <span><math><mi>I</mi><mo>:</mo><mo>=</mo><mo>〈</mo><mi>f</mi><mo>〉</mo></math></span> and <span><math><mi>B</mi><mo>:</mo><mo>=</mo><mi>k</mi><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo><mo>/</mo><mi>I</mi></math></span>. Let <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> be a positive integer. In this article, we compute the module of logarithmic differential operators along <em>I</em>, with order <em>n</em>, from the datum of the polynomial <em>f</em>. For <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span>, we show that our method produces a closed-form presentation of this module, which yields, as a by-product, an effective proof of the Nakai conjecture for the reduced (not necessarily irreducible) homogeneous plane curve singularities. As a theoretical cornerstone of this computation, we begin by proving that the order filtration on the (left) <em>B</em>-module of the differential operators, with order <em>n</em>, denoted by <span><math><msubsup><mrow><mi>Diff</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>(</mo><mi>B</mi><mo>)</mo></math></span>, actually defines a grading on <span><math><msubsup><mrow><mi>Diff</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>(</mo><mi>B</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"136 ","pages":"Article 102556"},"PeriodicalIF":1.1,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079987","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":"Sum-and-quotient characteristic decomposition of polynomial ideals","authors":"Dongming Wang ,&nbsp;Linpeng Wang","doi":"10.1016/j.jsc.2026.102553","DOIUrl":"10.1016/j.jsc.2026.102553","url":null,"abstract":"<div><div>This paper studies the algorithmic decomposition of polynomial ideals using the sum-and-quotient operation, a key technique implicitly involved in the process of computing Hilbert polynomials. We extend the framework of characteristic decomposition algorithms that maintain Hilbert polynomial relationships to the positive-dimensional case. In particular, we propose algorithms that successively apply the sum-and-quotient lemma to decompose any given polynomial ideal into finitely many ideals generated by regular sets or regular sequences, such that certain relations among the zero sets and the Hilbert polynomials are simultaneously preserved. This approach offers a new framework for representing the zero sets of polynomial ideals with multiplicities and reveals inherent connections among key concepts in the algorithmic theories of triangular sets, Gröbner bases, and Hilbert polynomials. We provide examples to illustrate computational properties and contrasts between our method and pseudo-division-based triangular decomposition algorithms. Experimental results demonstrate that the performance of our method is comparable with the existing triangular decomposition algorithms based solely on Gröbner bases computation.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"136 ","pages":"Article 102553"},"PeriodicalIF":1.1,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039594","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":"Secant varieties of Segre-Veronese varieties Pm×Pn embedded by O(1,2) are non-defective for n ≫ m3, m ≥ 3","authors":"Matěj Doležálek ,&nbsp;Nikhil Ken","doi":"10.1016/j.jsc.2025.102546","DOIUrl":"10.1016/j.jsc.2025.102546","url":null,"abstract":"<div><div>We prove that for any <span><math><mi>m</mi><mo>≥</mo><mn>3</mn></math></span>, <span><math><mi>n</mi><mo>≫</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, all secant varieties of the Segre-Veronese variety <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> have the expected dimension. This was already proved by Abo and Brambilla in the subabundant case, hence we focus on the superabundant case. We generalize an approach due to Brambilla and Ottaviani into a construction we call the <em>inductant</em>. With a combinatorial investigation of these constructions, the proof of non-defectivity reduces to checking a finite collection of base cases, which we verify using a computer-assisted proof.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102546"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145924777","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":"A field-theoretic view of unlabeled sensing","authors":"Hao Liang,&nbsp;Jingyu Lu,&nbsp;Manolis C. Tsakiris,&nbsp;Lihong Zhi","doi":"10.1016/j.jsc.2025.102542","DOIUrl":"10.1016/j.jsc.2025.102542","url":null,"abstract":"<div><div>Unlabeled sensing is the problem of solving a linear system of equations, where the right-hand-side vector is known only up to a permutation. In this work, we study fields of rational functions related to symmetric polynomials and their images under a linear projection of the variables; as a consequence, we establish that the solution to an <em>n</em>-dimensional unlabeled sensing problem with generic data can be obtained as the unique solution to a system of <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span> polynomial equations of degrees <span><math><mn>1</mn><mo>,</mo><mspace></mspace><mn>2</mn><mo>,</mo><mspace></mspace><mo>…</mo><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn></math></span> in <em>n</em> unknowns. Besides the new theoretical insights, this development offers the potential for scaling up algebraic unlabeled sensing algorithms.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102542"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145839833","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":"Algorithms for 2-solvable difference equations","authors":"Heba Bou KaedBey,&nbsp;Mark van Hoeij","doi":"10.1016/j.jsc.2025.102550","DOIUrl":"10.1016/j.jsc.2025.102550","url":null,"abstract":"<div><div>Our paper (<span><span>Bou KaedBey et al., 2024</span></span>) gave two algorithms for solving difference equations in terms of lower order equations: an algorithm for absolute factorization, and an algorithm for solving third order equations in terms of second order. Here we improve the efficiency for absolute factorization, and extend the other algorithm to order four.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102550"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145883879","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":"Arithmetic properties of partition functions introduced by Pushpa and Vasuki","authors":"Hemjyoti Nath ,&nbsp;Manjil P. Saikia","doi":"10.1016/j.jsc.2026.102555","DOIUrl":"10.1016/j.jsc.2026.102555","url":null,"abstract":"<div><div>In this short note, we prove several infinite families of congruences for certain restricted partition functions introduced by <span><span>Pushpa and Vasuki (2022)</span></span>, thereby also proving a conjecture of <span><span>Dasappa et al. (2024)</span></span>. We further establish some isolated congruences that appear to have been overlooked by earlier authors. Our proofs employ several methods: elementary dissections, Huffing operator technique, and nontrivial applications of the algorithmic method of Radu (and Smoot).</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102555"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976245","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":"A computational approach to rational summability and its applications via discrete residues","authors":"Carlos E. Arreche ,&nbsp;Hari P. Sitaula","doi":"10.1016/j.jsc.2025.102552","DOIUrl":"10.1016/j.jsc.2025.102552","url":null,"abstract":"<div><div>A rational function <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is rationally summable if there exists a rational function <span><math><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> such that <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>−</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. Detecting whether a given rational function is summable is an important and basic computational subproblem that arises in algorithms to study diverse aspects of shift difference equations. The discrete residues introduced by Chen and Singer in 2012 enjoy the obstruction-theoretic property that a rational function is summable if and only if all its discrete residues vanish. However, these discrete residues are defined in terms of the data in the complete partial fraction decomposition of the given rational function, which cannot be accessed computationally in general. We explain how to efficiently compute (a rational representation of) the discrete residues of any rational function, relying only on gcd computations, linear algebra, and a black box algorithm to compute the autodispersion set of the denominator polynomial. We also explain how to apply our algorithms to serial summability and creative telescoping problems, and how to apply these computations to compute Galois groups of difference equations.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102552"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145924778","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":"Superspecial genus-4 double covers of elliptic curves","authors":"Takumi Ogasawara ,&nbsp;Ryo Ohashi ,&nbsp;Kosuke Sakata ,&nbsp;Shushi Harashita","doi":"10.1016/j.jsc.2025.102548","DOIUrl":"10.1016/j.jsc.2025.102548","url":null,"abstract":"<div><div>In this paper we study genus-4 curves obtained as double covers of elliptic curves. Firstly we shall give explicit defining equations of such curves with explicit criterion for whether it is nonsingular, and show the irreducibility of the long polynomial determining whether the genus-4 curve is nonsingular or not, in any characteristic <span><math><mo>≠</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span>. Secondly, as an application, we enumerate superspecial genus-4 double covers of elliptic curves in small characteristic.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102548"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145924779","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 roadmaps in unbounded smooth real algebraic sets II: Algorithm and complexity","authors":"Rémi Prébet ,&nbsp;Mohab Safey El Din ,&nbsp;Éric Schost","doi":"10.1016/j.jsc.2025.102532","DOIUrl":"10.1016/j.jsc.2025.102532","url":null,"abstract":"<div><div>A roadmap for an algebraic set <em>V</em> defined by polynomials with coefficients in the field <span><math><mi>Q</mi></math></span> of rational numbers is an algebraic curve contained in <em>V</em> whose intersection with all connected components of <span><math><mi>V</mi><mo>∩</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is connected. These objects, introduced by Canny, can be used to answer connectivity queries over <span><math><mi>V</mi><mo>∩</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> provided that they are required to contain the finite set of query points <span><math><mi>P</mi><mo>⊂</mo><mi>V</mi></math></span>; in this case, we say that the roadmap is associated to <span><math><mo>(</mo><mi>V</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span>.</div><div>In this paper, we make effective a connectivity result we previously proved, to design a Monte Carlo algorithm which, on input <em>(i)</em> a finite sequence of polynomials defining <em>V</em> (and satisfying some regularity assumptions) and <em>(ii)</em> an algebraic representation of finitely many query points <span><math><mi>P</mi></math></span> in <em>V</em>, computes a roadmap for <span><math><mo>(</mo><mi>V</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span>. This algorithm generalizes the nearly optimal one introduced by the last two authors by dropping a boundedness assumption on the real trace of <em>V</em>.</div><div>The output size and running times of our algorithm are both polynomial in <span><math><msup><mrow><mo>(</mo><mi>n</mi><mi>D</mi><mo>)</mo></mrow><mrow><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>d</mi></mrow></msup></math></span>, where <em>D</em> is the maximal degree of the input equations and <em>d</em> is the dimension of <em>V</em>. As far as we know, the best previously known algorithm dealing with such sets has an output size and running time respectively polynomial in <span><math><msup><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow></msup><mi>D</mi><mo>)</mo></mrow><mrow><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow></msup></math></span> and <span><math><msup><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow></msup><mi>D</mi><mo>)</mo></mrow><mrow><mi>n</mi><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>n</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102532"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685319","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}