Journal of Symbolic Computation最新文献

Elimination by substitution

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2025-04-02 DOI: 10.1016/j.jsc.2025.102445

Martin Kreuzer , Lorenzo Robbiano

{"title":"Elimination by substitution","authors":"Martin Kreuzer , Lorenzo Robbiano","doi":"10.1016/j.jsc.2025.102445","DOIUrl":"10.1016/j.jsc.2025.102445","url":null,"abstract":"<div><div>Let <em>K</em> be a field and <span><math><mi>P</mi><mo>=</mo><mi>K</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>. The technique of elimination by substitution is based on discovering a coherently <span><math><mi>Z</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></math></span>-separating tuple of polynomials <span><math><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></math></span> in an ideal <em>I</em>, i.e., on finding polynomials such that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> with <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>K</mi><mo>[</mo><mi>X</mi><mo>∖</mo><mi>Z</mi><mo>]</mo></math></span>. Here we elaborate on this technique in the case when <em>P</em> is non-negatively graded. The existence of a coherently <em>Z</em>-separating tuple is reduced to solving several <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-module membership problems. Best separable re-embeddings, i.e., isomorphisms <span><math><mi>P</mi><mo>/</mo><mi>I</mi><mo>⟶</mo><mi>K</mi><mo>[</mo><mi>X</mi><mo>∖</mo><mi>Z</mi><mo>]</mo><mo>/</mo><mo>(</mo><mi>I</mi><mo>∩</mo><mi>K</mi><mo>[</mo><mi>X</mi><mo>∖</mo><mi>Z</mi><mo>]</mo><mo>)</mo></math></span> with maximal #<em>Z</em>, are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>⟶</mo><mi>P</mi><mo>/</mo><mi>I</mi></math></span> as a fibration over an affine space, we show that its fibers allow optimal <em>Z</em>-separating re-embeddings, and we provide a criterion for a fiber to be isomorphic to an affine space. In the last section we introduce a new technique based on the solution of a unimodular matrix problem which enables us to construct automorphisms of <em>P</em> such that additional <em>Z</em>-separating re-embeddings are possible. One of the main outcomes is an algorithm which allows us to explicitly compute a homogeneous isomorphism between <span><math><mi>P</mi><mo>/</mo><mi>I</mi></math></span> and a non-negatively graded polynomial ring if <span><math><mi>P</mi><mo>/</mo><mi>I</mi></math></span> is regular.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102445"},"PeriodicalIF":0.6,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768989","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}

引用次数: 0

Proof of some conjectural congruences involving products of two binomial coefficients

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2025-02-27 DOI: 10.1016/j.jsc.2025.102436

Guo-Shuai Mao , Xiran Zhang

引用次数: 0

Certified simultaneous isotopic approximation of algebraic curves via subdivision

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2025-02-26 DOI: 10.1016/j.jsc.2025.102435

Michael Burr, Michael Byrd

引用次数: 0

A propositional encoding for first-order clausal entailment over infinitely many constants

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2025-02-19 DOI: 10.1016/j.jsc.2025.102434

Vaishak Belle

{"title":"A propositional encoding for first-order clausal entailment over infinitely many constants","authors":"Vaishak Belle","doi":"10.1016/j.jsc.2025.102434","DOIUrl":"10.1016/j.jsc.2025.102434","url":null,"abstract":"<div><div>There is a fundamental trade-off between the expressiveness of the language and the tractability of the reasoning task in knowledge representation. On the one hand it is widely acknowledged that relations and more generally, the expressiveness of first-order logic is extremely useful for capturing concepts required for common-sense reasoning. But at the same time the entailment problem is only semi-decidable.</div><div>There have been a wide range of approaches to deal with this trade-off, from restricting the language to propositional logic to limit the expressiveness of the language in terms of the arity of the predicates (as in description logics) or the use of negation (as in Horn logic) to limit reasoning by weakening the entailment relation using non-standard semantics.</div><div>In this work, we address a gap in this literature. We show that there is an intuitive fragment of first-order disjunctive knowledge, for which reasoning is decidable and can be reduced to propositional satisfiability. Knowledge bases in this fragment correspond to universally quantified first-order clauses, but without arity restrictions and without restrictions on the appearance of negation. Queries, however, are expected to be ground formulas. We achieve this result by showing how the entailment over infinitely many infinite-sized structures can be reduced to a search over finitely many finite-size structures. The crux of the argument lies in showing that constants not mentioned in the knowledge base and/or query behave identically (in a suitable formal sense). We then go on to also show that there is also an extension to this result for function symbols.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102434"},"PeriodicalIF":0.6,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474775","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}

引用次数: 0

Reduction systems and degree bounds for integration

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2025-02-19 DOI: 10.1016/j.jsc.2025.102432

Hao Du , Clemens G. Raab

引用次数: 0

Congruence properties for Schmidt type d-fold partition diamonds

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2025-02-19 DOI: 10.1016/j.jsc.2025.102431

Olivia X.M. Yao, Xuan Yu

{"title":"Congruence properties for Schmidt type d-fold partition diamonds","authors":"Olivia X.M. Yao, Xuan Yu","doi":"10.1016/j.jsc.2025.102431","DOIUrl":"10.1016/j.jsc.2025.102431","url":null,"abstract":"<div><div>Recently, Dockery, Jameson, Sellers and Wilson introduced new combinatorial objects called <em>d</em>-fold partition diamonds, which generalize both the classical partition function and the plane partition diamonds of Andrews, Paule and Riese. They also investigated a partition function <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> which counts the number of Schmidt type <em>d</em>-fold partition diamonds of <em>n</em>. They presented the generating functions of <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and proved several congruences for <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. At the end of their paper, they posed a conjecture on congruences modulo 7 for <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. In this paper, we prove the conjectural congruences for <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> by using two methods: an elementary proof based on a result of Garvan and an algorithmic proof based on the Mathematica package RaduRK by Smoot. We also give an algorithmic proof of the conjectural congruences for <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> by utilizing Smoot's Mathematica package RaduRK. In addition, we prove new infinite families of congruences modulo 7 for <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and prove that <span><math><mfrac><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mn>7</mn><mi>n</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>7</mn></mrow></mfrac></math></span> takes integer values with probability 1 for <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>. Moreover, we show that there exist infinitely many integers <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> such that <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>12</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>≡</mo><mspace></mspace><mi>i</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>13</mn><mo>)</mo></math></span> with <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mn>12</mn></math></span>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102431"},"PeriodicalIF":0.6,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509114","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}

引用次数: 0

On other two representations of the C-recursive integer sequences by terms in modular arithmetic

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2025-02-19 DOI: 10.1016/j.jsc.2025.102433

Mihai Prunescu

引用次数: 0

Wilf-Zeilberger seeds and non-trivial hypergeometric identities

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2025-01-28 DOI: 10.1016/j.jsc.2025.102421

Kam Cheong Au

引用次数: 0

First-order factors of linear Mahler operators

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2025-01-28 DOI: 10.1016/j.jsc.2025.102424

Frédéric Chyzak , Thomas Dreyfus , Philippe Dumas , Marc Mezzarobba

引用次数: 0

The conjugacy problem and canonical representatives in finitely generated nilpotent groups

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2025-01-27 DOI: 10.1016/j.jsc.2025.102422

Bettina Eick, Óscar Fernández Ayala

{"title":"The conjugacy problem and canonical representatives in finitely generated nilpotent groups","authors":"Bettina Eick, Óscar Fernández Ayala","doi":"10.1016/j.jsc.2025.102422","DOIUrl":"10.1016/j.jsc.2025.102422","url":null,"abstract":"<div><div>We introduce a variation on the conjugacy problem for elements and subgroups in a finitely generated nilpotent group <em>G</em> given by a nilpotent presentation and we describe effective algorithms for its solution. While the classical conjugacy problem takes elements or subgroups <em>a</em> and <em>b</em> of <em>G</em> and asks to construct <span><math><mi>g</mi><mo>∈</mo><mi>G</mi></math></span> with <span><math><msup><mrow><mi>a</mi></mrow><mrow><mi>g</mi></mrow></msup><mo>=</mo><mi>b</mi></math></span>, our variation defines and determines a <em>canonical representative</em> <span><math><mi>C</mi><mi>a</mi><mi>n</mi><msub><mrow><mi>o</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>)</mo></math></span> in <span><math><msup><mrow><mi>a</mi></mrow><mrow><mi>G</mi></mrow></msup></math></span>. This allows to solve the conjugacy problem via an equality test <span><math><mi>C</mi><mi>a</mi><mi>n</mi><msub><mrow><mi>o</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><mi>C</mi><mi>a</mi><mi>n</mi><msub><mrow><mi>o</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>b</mi><mo>)</mo></math></span>. Additionally, our algorithms compute the associated centralizers or normalizers, respectively. We exhibit a variety of examples to demonstrate that our new methods are highly effective and often outperform the existing methods to solve the conjugacy problems for elements and subgroups in finitely generated nilpotent groups.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102422"},"PeriodicalIF":0.6,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176022","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}

引用次数: 0