Pub Date : 2025-02-19DOI: 10.1016/j.jsc.2025.102433
Mihai Prunescu
<div><div>If <span><math><mi>s</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> is a sequence satisfying a recurrence rule of the form:<span><span><span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>+</mo><mi>d</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>s</mi><mo>(</mo><mi>n</mi><mo>+</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>d</mi></mrow></msub><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span></span></span> with coefficients <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>Z</mi></math></span>, then there exist <span><math><mi>b</mi><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>N</mi></math></span> such that for all <span><math><mi>n</mi><mo>≥</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> the following representations work:<span><span><span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mrow><mo>⌊</mo><mfrac><mrow><mo>[</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo>)</mo><mo>+</mo><mo>⌈</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌉</mo></mrow></msup><mo>+</mo><mi>A</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>]</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>B</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>b</mi></mrow><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>n</mi></mrow></msup></mrow></mfrac><mo>⌋</mo></mrow><mo>,</mo></math></span></span></span><span><span><span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mo>|</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>|</mo></mrow></mfrac><mrow><mo>{</mo><mrow><mo>[</mo><mrow><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mrow><mo>⌈</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌉</mo></mrow></mrow></msup><mo>−</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msup><mi>sgn</mi><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mi>A</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow></mrow></mrow><mspace></mspace><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>B</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>]</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>}</mo><mo>.</mo></math></span></span></span> Here <span><math><mi>A</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>B</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> are polynomials with integer coefficients in <span><math><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Th
{"title":"On other two representations of the C-recursive integer sequences by terms in modular arithmetic","authors":"Mihai Prunescu","doi":"10.1016/j.jsc.2025.102433","DOIUrl":"10.1016/j.jsc.2025.102433","url":null,"abstract":"<div><div>If <span><math><mi>s</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> is a sequence satisfying a recurrence rule of the form:<span><span><span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>+</mo><mi>d</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>s</mi><mo>(</mo><mi>n</mi><mo>+</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>d</mi></mrow></msub><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span></span></span> with coefficients <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>Z</mi></math></span>, then there exist <span><math><mi>b</mi><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>N</mi></math></span> such that for all <span><math><mi>n</mi><mo>≥</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> the following representations work:<span><span><span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mrow><mo>⌊</mo><mfrac><mrow><mo>[</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo>)</mo><mo>+</mo><mo>⌈</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌉</mo></mrow></msup><mo>+</mo><mi>A</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>]</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>B</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>b</mi></mrow><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>n</mi></mrow></msup></mrow></mfrac><mo>⌋</mo></mrow><mo>,</mo></math></span></span></span><span><span><span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mo>|</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>|</mo></mrow></mfrac><mrow><mo>{</mo><mrow><mo>[</mo><mrow><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mrow><mo>⌈</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌉</mo></mrow></mrow></msup><mo>−</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msup><mi>sgn</mi><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mi>A</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow></mrow></mrow><mspace></mspace><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>B</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>]</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>}</mo><mo>.</mo></math></span></span></span> Here <span><math><mi>A</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>B</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> are polynomials with integer coefficients in <span><math><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Th","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102433"},"PeriodicalIF":0.6,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143454136","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 : 2025-01-28DOI: 10.1016/j.jsc.2025.102421
Kam Cheong Au
We introduce a systematic approach for generating Wilf-Zeilberger-pairs, and prove some hypergeometric identities conjectured by J. Guillera, Z.W. Sun, Y. Zhao and others, including two Ramanujan-, one formulas as well as a remarkable series for .
我们介绍了一种生成wilf - zeilberger对的系统方法，并证明了J. Guillera， Z.W. Sun， Y. Zhao等人猜想的一些超几何恒等式，包括两个Ramanujan-1/π4，一个1/π3公式以及ζ(5)的一个显著级数。
{"title":"Wilf-Zeilberger seeds and non-trivial hypergeometric identities","authors":"Kam Cheong Au","doi":"10.1016/j.jsc.2025.102421","DOIUrl":"10.1016/j.jsc.2025.102421","url":null,"abstract":"<div><div>We introduce a systematic approach for generating Wilf-Zeilberger-pairs, and prove some hypergeometric identities conjectured by J. Guillera, Z.W. Sun, Y. Zhao and others, including two Ramanujan-<span><math><mn>1</mn><mo>/</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>, one <span><math><mn>1</mn><mo>/</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> formulas as well as a remarkable series for <span><math><mi>ζ</mi><mo>(</mo><mn>5</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102421"},"PeriodicalIF":0.6,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176023","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 : 2025-01-28DOI: 10.1016/j.jsc.2025.102424
Frédéric Chyzak , Thomas Dreyfus , Philippe Dumas , Marc Mezzarobba
We develop and compare two algorithms for computing first-order right-hand factors in the ring of linear Mahler operators where are polynomials in x and for some integer . In other words, we give algorithms for finding all formal infinite product solutions of linear functional equations .
The first of our algorithms is adapted from Petkovšek's classical algorithm for the analogous problem in the case of linear recurrences. The second one proceeds by computing a basis of generalized power series solutions of the functional equation and by using Hermite–Padé approximants to detect those linear combinations of the solutions that correspond to first-order factors.
We present implementations of both algorithms and discuss their use in combination with criteria from the literature to prove the differential transcendence of power series solutions of Mahler equations.
{"title":"First-order factors of linear Mahler operators","authors":"Frédéric Chyzak , Thomas Dreyfus , Philippe Dumas , Marc Mezzarobba","doi":"10.1016/j.jsc.2025.102424","DOIUrl":"10.1016/j.jsc.2025.102424","url":null,"abstract":"<div><div>We develop and compare two algorithms for computing first-order right-hand factors in the ring of linear Mahler operators <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msub><msup><mrow><mi>M</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>M</mi><mo>+</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> where <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> are polynomials in <em>x</em> and <span><math><mi>M</mi><mi>x</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>b</mi></mrow></msup><mi>M</mi></math></span> for some integer <span><math><mi>b</mi><mo>≥</mo><mn>2</mn></math></span>. In other words, we give algorithms for finding all formal infinite product solutions of linear functional equations <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>b</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msup><mo>)</mo><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>b</mi></mrow></msup><mo>)</mo><mo>+</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>.</div><div>The first of our algorithms is adapted from Petkovšek's classical algorithm for the analogous problem in the case of linear recurrences. The second one proceeds by computing a basis of generalized power series solutions of the functional equation and by using Hermite–Padé approximants to detect those linear combinations of the solutions that correspond to first-order factors.</div><div>We present implementations of both algorithms and discuss their use in combination with criteria from the literature to prove the differential transcendence of power series solutions of Mahler equations.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102424"},"PeriodicalIF":0.6,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143288038","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 : 2025-01-27DOI: 10.1016/j.jsc.2025.102422
Bettina Eick, Óscar Fernández Ayala
We introduce a variation on the conjugacy problem for elements and subgroups in a finitely generated nilpotent group G given by a nilpotent presentation and we describe effective algorithms for its solution. While the classical conjugacy problem takes elements or subgroups a and b of G and asks to construct with , our variation defines and determines a canonical representative in . This allows to solve the conjugacy problem via an equality test . 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.
{"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}
Pub Date : 2025-01-27DOI: 10.1016/j.jsc.2025.102423
Lixin Du
We extend the criterion on the existence of telescopers for hypergeometric terms to the case of P-recursive sequences. This criterion is based on the concept of integral bases and the generalized Abramov-Petkovšek reduction for P-recursive sequences.
{"title":"On the existence of telescopers for P-recursive sequences","authors":"Lixin Du","doi":"10.1016/j.jsc.2025.102423","DOIUrl":"10.1016/j.jsc.2025.102423","url":null,"abstract":"<div><div>We extend the criterion on the existence of telescopers for hypergeometric terms to the case of P-recursive sequences. This criterion is based on the concept of integral bases and the generalized Abramov-Petkovšek reduction for P-recursive sequences.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102423"},"PeriodicalIF":0.6,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143177145","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 : 2025-01-27DOI: 10.1016/j.jsc.2025.102420
Guoce Xin , Yingrui Zhang , ZiHao Zhang
The Todd polynomials, denoted as , are characterized by their generating function: These polynomials serve as fundamental components in the Todd class of toric varieties – a concept of significant relevance in the study of lattice polytopes and number theory. We identify that generalized Todd polynomials emerge naturally within the realm of MacMahon's partition analysis, particularly in the context of computing the Ehrhart series. We introduce an efficient method for the evaluation of generalized Todd polynomials for numerical values of . This is achieved through the development of expedited operations in the quotient ring modulo , where p is a large prime. The practical implications of our work are demonstrated through two applications: firstly, we facilitate a recalculated resolution of the Ehrhart series for magic squares of order 6, a problem initially addressed by the first author, reducing computation time from 70 days to approximately 1 day; secondly, we present a polynomial-time algorithm for Integer Linear Programming in the scenario where the dimension is fixed, exhibiting a notable enhancement in computational efficiency.
{"title":"Fast evaluation of generalized Todd polynomials: Applications to MacMahon's partition analysis and integer programming","authors":"Guoce Xin , Yingrui Zhang , ZiHao Zhang","doi":"10.1016/j.jsc.2025.102420","DOIUrl":"10.1016/j.jsc.2025.102420","url":null,"abstract":"<div><div>The Todd polynomials, denoted as <span><math><msub><mrow><mtext>td</mtext></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span>, are characterized by their generating function:<span><span><span><math><munder><mo>∑</mo><mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></munder><msub><mrow><mtext>td</mtext></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><mfrac><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>s</mi></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>s</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow></mfrac><mo>.</mo></math></span></span></span> These polynomials serve as fundamental components in the Todd class of toric varieties – a concept of significant relevance in the study of lattice polytopes and number theory. We identify that generalized Todd polynomials emerge naturally within the realm of MacMahon's partition analysis, particularly in the context of computing the Ehrhart series. We introduce an efficient method for the evaluation of generalized Todd polynomials for numerical values of <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. This is achieved through the development of expedited operations in the quotient ring <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>[</mo><mo>[</mo><mi>s</mi><mo>]</mo><mo>]</mo></math></span> modulo <span><math><msup><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, where <em>p</em> is a large prime. The practical implications of our work are demonstrated through two applications: firstly, we facilitate a recalculated resolution of the Ehrhart series for magic squares of order 6, a problem initially addressed by the first author, reducing computation time from 70 days to approximately 1 day; secondly, we present a polynomial-time algorithm for Integer Linear Programming in the scenario where the dimension is fixed, exhibiting a notable enhancement in computational efficiency.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102420"},"PeriodicalIF":0.6,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176021","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 : 2025-01-13DOI: 10.1016/j.jsc.2025.102419
Heba Bou KaedBey, Mark van Hoeij, Man Cheung Tsui
We classify order 3 linear difference operators over that are solvable in terms of lower order difference operators. To prove this result, we introduce the notion of absolute irreducibility for difference modules, and classify modules of arbitrary dimension that are irreducible but not absolutely irreducible.
{"title":"Solving order 3 difference equations","authors":"Heba Bou KaedBey, Mark van Hoeij, Man Cheung Tsui","doi":"10.1016/j.jsc.2025.102419","DOIUrl":"10.1016/j.jsc.2025.102419","url":null,"abstract":"<div><div>We classify order 3 linear difference operators over <span><math><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> that are solvable in terms of lower order difference operators. To prove this result, we introduce the notion of absolute irreducibility for difference modules, and classify modules of arbitrary dimension that are irreducible but not absolutely irreducible.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102419"},"PeriodicalIF":0.6,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159664","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 : 2024-12-17DOI: 10.1016/j.jsc.2024.102418
Carlos D'Andrea, Hoon Hong, Evelyne Hubert, Teresa Krick
{"title":"Computational Algebra and Geometry: A special issue in memory and honor of Agnes Szanto","authors":"Carlos D'Andrea, Hoon Hong, Evelyne Hubert, Teresa Krick","doi":"10.1016/j.jsc.2024.102418","DOIUrl":"10.1016/j.jsc.2024.102418","url":null,"abstract":"","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102418"},"PeriodicalIF":0.6,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160245","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 : 2024-12-09DOI: 10.1016/j.jsc.2024.102417
Alexander Ushakov, Chloe Weiers
In this paper we study the complexity of solving quadratic equations in the lamplighter group. We give a complete classification of cases (depending on genus and other characteristics of a given equation) when the problem is NP-complete or polynomial-time decidable. We notice that the conjugacy problem can be solved in linear time. Finally, we prove that the problem belongs to the class XP.
{"title":"Quadratic equations in the lamplighter group","authors":"Alexander Ushakov, Chloe Weiers","doi":"10.1016/j.jsc.2024.102417","DOIUrl":"10.1016/j.jsc.2024.102417","url":null,"abstract":"<div><div>In this paper we study the complexity of solving quadratic equations in the lamplighter group. We give a complete classification of cases (depending on genus and other characteristics of a given equation) when the problem is <strong>NP</strong>-complete or polynomial-time decidable. We notice that the conjugacy problem can be solved in linear time. Finally, we prove that the problem belongs to the class <strong>XP</strong>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102417"},"PeriodicalIF":0.6,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128507","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 : 2024-12-04DOI: 10.1016/j.jsc.2024.102416
Cyrille Chenavier, Thomas Cluzeau, Adya Musson-Leymarie
We investigate formal power series ideals and their relationship to topological rewriting theory. Since commutative formal power series algebras are Zariski rings, their ideals are closed for the adic topology defined by the maximal ideal generated by the indeterminates. We provide a constructive proof of this result which, given a formal power series in the topological closure of an ideal, consists in computing a cofactor representation of the series with respect to a standard basis of the ideal. We apply this result in the context of topological rewriting theory, where two natural notions of confluence arise: topological confluence and infinitary confluence. We give explicit examples illustrating that in general, infinitary confluence is a strictly stronger notion than topological confluence. Using topological closure of ideals, we finally show that in the context of rewriting theory on commutative formal power series, infinitary and topological confluences are equivalent when the monomial order considered is compatible with the degree.
{"title":"Topological closure of formal powers series ideals and application to topological rewriting theory","authors":"Cyrille Chenavier, Thomas Cluzeau, Adya Musson-Leymarie","doi":"10.1016/j.jsc.2024.102416","DOIUrl":"10.1016/j.jsc.2024.102416","url":null,"abstract":"<div><div>We investigate formal power series ideals and their relationship to topological rewriting theory. Since commutative formal power series algebras are Zariski rings, their ideals are closed for the adic topology defined by the maximal ideal generated by the indeterminates. We provide a constructive proof of this result which, given a formal power series in the topological closure of an ideal, consists in computing a cofactor representation of the series with respect to a standard basis of the ideal. We apply this result in the context of topological rewriting theory, where two natural notions of confluence arise: topological confluence and infinitary confluence. We give explicit examples illustrating that in general, infinitary confluence is a strictly stronger notion than topological confluence. Using topological closure of ideals, we finally show that in the context of rewriting theory on commutative formal power series, infinitary and topological confluences are equivalent when the monomial order considered is compatible with the degree.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102416"},"PeriodicalIF":0.6,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159654","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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