Pub Date : 2026-01-26DOI: 10.1016/j.jsc.2026.102556
Julien Sebag
Let k be a field with characteristic zero. Let be a reduced homogeneous polynomial with degree . We set and . Let be a positive integer. In this article, we compute the module of logarithmic differential operators along I, with order n, from the datum of the polynomial f. For , 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) B-module of the differential operators, with order n, denoted by , actually defines a grading on .
设k为特征为零的场。设f∈k[x，y]为阶数d≥1的简化齐次多项式。设I:= < f >, B:=k[x,y]/I。设n≥2为正整数。在本文中，我们从多项式f的基准计算沿I的n阶对数微分算子的模。对于n=2，我们证明了我们的方法产生了该模的封闭形式表示，作为副产品，它产生了关于可约（不一定不可约）齐次平面曲线奇点的Nakai猜想的有效证明。作为这个计算的理论基础，我们首先证明微分算子的（左）B模上的阶过滤，用Diffkn(B)表示为n阶，实际上定义了Diffkn(B)上的分级。
{"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":"2026-01-26","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}
Pub Date : 2026-01-13DOI: 10.1016/j.jsc.2026.102554
Jutta Rath, Roswitha Rissner
It is known that for a monomial ideal I, the number of minimal generators, , eventually follows a polynomial pattern for increasing n. 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 , where d is a constant bounded above by the maximal x- or y-degree appearing in the set of minimal generators of I. We show that every higher power for any can be constructed from certain subideals of . This provides an explicit description of in terms of . Given , this construction significantly reduces computational complexity in determining larger powers of I. This further enables us to explicitly compute for all in terms of a linear polynomial in n. We include runtime measurements for the attached implementation in SageMath.
{"title":"Minimal generating sets of large powers of bivariate monomial ideals","authors":"Jutta Rath, 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":"2026-01-13","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}
Pub Date : 2026-01-13DOI: 10.1016/j.jsc.2026.102553
Dongming Wang , Linpeng Wang
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.
{"title":"Sum-and-quotient characteristic decomposition of polynomial ideals","authors":"Dongming Wang , 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":"2026-01-13","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}
Pub Date : 2026-01-12DOI: 10.1016/j.jsc.2026.102555
Hemjyoti Nath , Manjil P. Saikia
In this short note, we prove several infinite families of congruences for certain restricted partition functions introduced by Pushpa and Vasuki (2022), thereby also proving a conjecture of Dasappa et al. (2024). 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).
{"title":"Arithmetic properties of partition functions introduced by Pushpa and Vasuki","authors":"Hemjyoti Nath , 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":"2026-01-12","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}
Pub Date : 2025-12-31DOI: 10.1016/j.jsc.2025.102552
Carlos E. Arreche , Hari P. Sitaula
A rational function is rationally summable if there exists a rational function such that . 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.
{"title":"A computational approach to rational summability and its applications via discrete residues","authors":"Carlos E. Arreche , 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":"2025-12-31","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}
Loop invariants are properties of a program loop that hold both before and after each iteration of the loop. They are often used to verify programs and ensure that algorithms consistently produce correct results during execution. Consequently, generating invariants becomes a crucial task for loops. We specifically focus on polynomial loops, where both the loop conditions and the assignments within the loop are expressed as polynomials. Although computing polynomial invariants for general loops is undecidable, efficient algorithms have been developed for certain classes of loops. For instance, when all assignments within a while loop involve linear polynomials, the loop becomes solvable. In this work, we study the more general case, where the polynomials can have arbitrary degrees.
Using tools from algebraic geometry, we present two algorithms designed to generate all polynomial invariants within a given vector subspace, for a branching loop with nondeterministic conditional statements. These algorithms combine linear algebraic subroutines with computations on polynomial ideals. They differ depending on whether the initial values of the loop variables are specified or treated as parameters. Additionally, we present a much more efficient algorithm for generating polynomial invariants of a specific form, applicable to all initial values. This algorithm avoids expensive ideal computations.
{"title":"Algebraic and algorithmic methods for computing polynomial loop invariants","authors":"Erdenebayar Bayarmagnai , Fatemeh Mohammadi , Rémi Prébet","doi":"10.1016/j.jsc.2025.102551","DOIUrl":"10.1016/j.jsc.2025.102551","url":null,"abstract":"<div><div>Loop invariants are properties of a program loop that hold both before and after each iteration of the loop. They are often used to verify programs and ensure that algorithms consistently produce correct results during execution. Consequently, generating invariants becomes a crucial task for loops. We specifically focus on polynomial loops, where both the loop conditions and the assignments within the loop are expressed as polynomials. Although computing polynomial invariants for general loops is undecidable, efficient algorithms have been developed for certain classes of loops. For instance, when all assignments within a while loop involve linear polynomials, the loop becomes solvable. In this work, we study the more general case, where the polynomials can have arbitrary degrees.</div><div>Using tools from algebraic geometry, we present two algorithms designed to generate all polynomial invariants within a given vector subspace, for a branching loop with nondeterministic conditional statements. These algorithms combine linear algebraic subroutines with computations on polynomial ideals. They differ depending on whether the initial values of the loop variables are specified or treated as parameters. Additionally, we present a much more efficient algorithm for generating polynomial invariants of a specific form, applicable to all initial values. This algorithm avoids expensive ideal computations.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102551"},"PeriodicalIF":1.1,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145924766","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 : 2025-12-30DOI: 10.1016/j.jsc.2025.102550
Heba Bou KaedBey, Mark van Hoeij
Our paper (Bou KaedBey et al., 2024) 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.
我们的论文（Bou KaedBey et al., 2024）给出了两种用低阶方程求解差分方程的算法：一种是绝对分解算法，另一种是用二阶方程求解三阶方程的算法。本文提高了绝对分解的效率，并将另一种算法推广到4阶。
{"title":"Algorithms for 2-solvable difference equations","authors":"Heba Bou KaedBey, 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":"2025-12-30","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}
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 . Secondly, as an application, we enumerate superspecial genus-4 double covers of elliptic curves in small characteristic.
{"title":"Superspecial genus-4 double covers of elliptic curves","authors":"Takumi Ogasawara , Ryo Ohashi , Kosuke Sakata , 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":"2025-12-30","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}
Pub Date : 2025-12-30DOI: 10.1016/j.jsc.2025.102549
Zhaoxing Qi , Chenqi Mou
Triangular decomposition is a versatile computational tool for studying polynomial ideals, but the complexity is not well understood due to its intricate behaviors. Inspired by the recent interplay between polynomial system solving and chordal graphs, in this paper we analyze the complexity of triangular decomposition by using chordal graphs.
We first introduce a new vertex ordering of graphs called the substrong elimination one which characterizes strongly chordal graphs. Using this ordering, we propose a polynomial selection strategy for triangular decomposition over and show that for polynomial sets with strongly chordal associated graphs, the variables of any polynomial occurring in the decomposition with this strategy are contained in certain maximal clique. For ℓ polynomials in n variables with such an associated graph of treewidth m, under certain worst-case assumptions, the complexity of triangular decomposition over using this strategy is proved to be , smaller than the original when .
To extend these results to other algorithms for triangular decomposition over any field, we introduce the concept of transitive chordal graph based on the transitive vertex ordering. Then when the transitive perfect elimination ordering is used for a collection of algorithms, the similar inclusion of the variables in the maximal cliques is proved without requiring additional selection strategies. As a direct consequence, these algorithms are chordality-preserving when combined with transitive chordal graphs. At the end, we provide complexity analyses for two algorithms using transitive chordal graphs of bounded treewidths.
{"title":"Strongly and transitive chordal graphs and their applications in complexity analysis of triangular decomposition","authors":"Zhaoxing Qi , Chenqi Mou","doi":"10.1016/j.jsc.2025.102549","DOIUrl":"10.1016/j.jsc.2025.102549","url":null,"abstract":"<div><div>Triangular decomposition is a versatile computational tool for studying polynomial ideals, but the complexity is not well understood due to its intricate behaviors. Inspired by the recent interplay between polynomial system solving and chordal graphs, in this paper we analyze the complexity of triangular decomposition by using chordal graphs.</div><div>We first introduce a new vertex ordering of graphs called the substrong elimination one which characterizes strongly chordal graphs. Using this ordering, we propose a polynomial selection strategy for triangular decomposition over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and show that for polynomial sets with strongly chordal associated graphs, the variables of any polynomial occurring in the decomposition with this strategy are contained in certain maximal clique. For <em>ℓ</em> polynomials in <em>n</em> variables with such an associated graph of treewidth <em>m</em>, under certain worst-case assumptions, the complexity of triangular decomposition over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> using this strategy is proved to be <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mn>4</mn></mrow><mrow><mi>m</mi></mrow></msup><mi>ℓ</mi><mi>n</mi><msup><mrow><mo>(</mo><mfrac><mrow><mi>m</mi><mi>ℓ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></math></span>, smaller than the original <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> when <span><math><mi>m</mi><mo>≪</mo><mi>n</mi></math></span>.</div><div>To extend these results to other algorithms for triangular decomposition over any field, we introduce the concept of transitive chordal graph based on the transitive vertex ordering. Then when the transitive perfect elimination ordering is used for a collection of algorithms, the similar inclusion of the variables in the maximal cliques is proved without requiring additional selection strategies. As a direct consequence, these algorithms are chordality-preserving when combined with transitive chordal graphs. At the end, we provide complexity analyses for two algorithms using transitive chordal graphs of bounded treewidths.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102549"},"PeriodicalIF":1.1,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145924764","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 : 2025-12-29DOI: 10.1016/j.jsc.2025.102547
Boulos El Hilany , Martin Helmer , Elias Tsigaridas
We define a polyhedral version of a stratification for projection maps that applies to any complex or real toric variety and show that it yields similarly desirable properties to the classical map stratification of a proper map. Our results are constructive and give rise to a method for associating the Whitney strata of the projection to the faces of the polytope of the corresponding toric variety. For all the examples we consider, our resulting algorithm outperforms known general purpose methods, e.g., Helmer and Nanda (FoCM, 2022), and Đinh and Jelonek (DCG, 2021), for computing map stratifications.
{"title":"Stratification of projection maps from toric varieties","authors":"Boulos El Hilany , Martin Helmer , Elias Tsigaridas","doi":"10.1016/j.jsc.2025.102547","DOIUrl":"10.1016/j.jsc.2025.102547","url":null,"abstract":"<div><div>We define a polyhedral version of a stratification for projection maps that applies to any complex or real toric variety and show that it yields similarly desirable properties to the classical map stratification of a proper map. Our results are constructive and give rise to a method for associating the Whitney strata of the projection to the faces of the polytope of the corresponding toric variety. For all the examples we consider, our resulting algorithm outperforms known general purpose methods, e.g., Helmer and Nanda (FoCM, 2022), and Đinh and Jelonek (DCG, 2021), for computing map stratifications.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102547"},"PeriodicalIF":1.1,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145883344","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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