Journal of Computational Algebra最新文献

{"title":"Nested fifth root radical identities from elliptic curves","authors":"Joseph Tonien","doi":"10.1016/j.jaca.2025.100032","DOIUrl":"10.1016/j.jaca.2025.100032","url":null,"abstract":"<div><div>Ramanujan discovered the following elegant identity involving cube roots:<span><span><span><math><msqrt><mrow><mi>m</mi><mroot><mrow><mn>4</mn><mo>(</mo><mi>m</mi><mo>−</mo><mn>2</mn><mi>n</mi><mo>)</mo></mrow><mrow><mn>3</mn></mrow></mroot><mo>+</mo><mi>n</mi><mroot><mrow><mn>4</mn><mi>m</mi><mo>+</mo><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mroot></mrow></msqrt><mo>=</mo><mo>±</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>(</mo><mroot><mrow><msup><mrow><mo>(</mo><mn>4</mn><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>3</mn></mrow></mroot><mo>+</mo><mroot><mrow><mn>4</mn><mo>(</mo><mi>m</mi><mo>−</mo><mn>2</mn><mi>n</mi><mo>)</mo><mo>(</mo><mn>4</mn><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>3</mn></mrow></mroot><mo>−</mo><mroot><mrow><mn>2</mn><msup><mrow><mo>(</mo><mi>m</mi><mo>−</mo><mn>2</mn><mi>n</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>3</mn></mrow></mroot><mo>)</mo><mo>.</mo></math></span></span></span></div><div>The goal of this paper is to derive nested fifth root radical identities in the form of<span><span><span><math><msqrt><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mroot><mrow><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mn>5</mn></mrow></mroot><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mroot><mrow><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>5</mn></mrow></mroot></mrow></msqrt><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mroot><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mn>5</mn></mrow></mroot><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub><mroot><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>5</mn></mrow></mroot><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub><mroot><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mrow><mn>5</mn></mrow></mroot><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>4</mn></mrow></msub><mroot><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow><mrow><mn>5</mn></mrow></mroot><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>5</mn></mrow></msub><mroot><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow><mrow><mn>5</mn></mrow></mroot><mo>.</mo></math></span></span></span> We show that these identities can be derived from a specific family of elliptic curves. Furthermore, we include the SageMath code for computing these expressions.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"13 ","pages":"Article 100032"},"PeriodicalIF":0.0,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ideals of generic forms","authors":"Ralf Fröberg","doi":"10.1016/j.jaca.2025.100033","DOIUrl":"10.1016/j.jaca.2025.100033","url":null,"abstract":"<div><div>We determine the Hilbert series of some classes of ideals generated by generic forms of degree two and three, and investigate the difference to the Hilbert series of ideals generated by powers of linear generic forms of the corresponding degrees.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"13 ","pages":"Article 100033"},"PeriodicalIF":0.0,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143844529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computational and theoretical aspects of rational parametrization of generalized tubular surfaces","authors":"J. William Hoffman ,&nbsp;Haohao Wang","doi":"10.1016/j.jaca.2025.100030","DOIUrl":"10.1016/j.jaca.2025.100030","url":null,"abstract":"<div><div>This paper consists of two components - a computational part and a theoretical part. The former targets the computer-aided geometric design of tubular surfaces. The latter focuses on the algebraic geometry of a family of conic curves. At the application level, we provide a straightforward and easy to implement computational algorithm to rationally parametrize generalized real tubular surfaces via moving lines. We discover that syzygies, i.e., moving lines, can be calculated directly from a given implicit equation of a projective conic. Specifically, we describe two linear polynomial vectors in 3-space whose entries are formulated in terms of the coefficients of the given implicit equation of the conic. We then prove that these two vectors are, in fact, a <em>μ</em>-basis, the generators for the syzygy module of the given conic, and furnish the rational parametrization of the given conic. At the theoretical level, we first briefly review the classical projection method for a rational parametrization of a generic non-degenerate conic. This is compared to the syzygy method, i.e., moving lines. We conclude the paper with an illustrative figure that depicts and compares the classical projection method and our moving line method.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"13 ","pages":"Article 100030"},"PeriodicalIF":0.0,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Zeros of S-characters","authors":"Thomas Breuer ,&nbsp;Michael Joswig ,&nbsp;Gunter Malle","doi":"10.1016/j.jaca.2025.100031","DOIUrl":"10.1016/j.jaca.2025.100031","url":null,"abstract":"<div><div>The concept of <em>S</em>-characters of finite groups was introduced by Zhmud' as a generalisation of transitive permutation characters. Any non-trivial <em>S</em>-character takes a zero value on some group element. By a deep result depending on the classification of finite simple groups a non-trivial transitive permutation character even vanishes on some element of prime power order. J-P. Serre asked whether this generalises to <em>S</em>-characters. We provide a translation of this question into the language of polyhedral geometry and thereby construct many counterexamples, using the capabilities of the computer algebra system <span>OSCAR</span>.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"13 ","pages":"Article 100031"},"PeriodicalIF":0.0,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143767461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Symmetric perfect 2-colorings of J(10,3)","authors":"Paul Tricot","doi":"10.1016/j.jaca.2024.100028","DOIUrl":"10.1016/j.jaca.2024.100028","url":null,"abstract":"","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"12 ","pages":"Article 100028"},"PeriodicalIF":0.0,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142700264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Improved computation of polynomial roots over number fields when using complex embeddings","authors":"Andrea Lesavourey ,&nbsp;Thomas Plantard ,&nbsp;Willy Susilo","doi":"10.1016/j.jaca.2024.100026","DOIUrl":"10.1016/j.jaca.2024.100026","url":null,"abstract":"<div><div>We explore a fairly generic method to compute roots of polynomials over number fields through complex embeddings. Our main contribution is to show how to use a structure of a relative extension to decode in a subfield. Additionally we describe several heuristic options to improve practical efficiency. We provide experimental data from our implementation and compare our methods to the state of the art algorithm implemented in <span>Pari/Gp</span>.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"12 ","pages":"Article 100026"},"PeriodicalIF":0.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142538675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Signature-based algorithm under non-compatible term orders and its application to change of ordering","authors":"Masayuki Noro","doi":"10.1016/j.jaca.2024.100027","DOIUrl":"10.1016/j.jaca.2024.100027","url":null,"abstract":"<div><div>The notion of the compatibility between a term order in a polynomial ring <em>R</em> and a module term order in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>l</mi></mrow></msup></math></span> is crucial to ensure the termination of a signature-based algorithm for general input ideals. However, it is shown experimentally that the compatibility does not necessarily imply efficient computation. Our experiments show that combining non-compatible term orders can improve performance for computing Gröbner bases with respect to some term orders. In such cases, we can use the Hilbert function to guarantee the termination. The Hilbert function can be computed by using a Gröbner basis with respect to some term order and thus the resulting algorithm is considered a change of ordering algorithm. In this paper, we give the details of the new change of ordering algorithm and we compare its performance with that of the usual Hilbert-driven Buchberger algorithm and the Gröbner walk algorithm.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"12 ","pages":"Article 100027"},"PeriodicalIF":0.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rational Askey–Wilson Bernstein bases and a multirational Askey–Wilson blossom","authors":"Plamen Simeonov ,&nbsp;Ron Goldman","doi":"10.1016/j.jaca.2024.100025","DOIUrl":"10.1016/j.jaca.2024.100025","url":null,"abstract":"<div><div>We introduce and study the properties of new negative degree rational Bernstein bases associated with the Askey–Wilson operator and we use these bases to define new types of rational Bernstein-Bézier curves. We also introduce a new type of blossom, the <em>multirational Askey–Wilson blossom</em>. We prove that four axioms uniquely characterize this blossom and we provide an explicit formula for this multirational blossom involving a right inverse of the Askey–Wilson operator. A formula for the coefficients of a function expanded in a rational Askey–Wilson Bernstein basis in terms of certain values of the Askey–Wilson operator is derived. We also establish a dual functional property that expresses the coefficients of these new types of rational Bernstein–Bézier curves in terms of values of their multirational Askey–Wilson blossom.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"12 ","pages":"Article 100025"},"PeriodicalIF":0.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Factoring perfect reconstruction filter banks into causal lifting matrices: A Diophantine approach","authors":"Christopher M. Brislawn","doi":"10.1016/j.jaca.2024.100024","DOIUrl":"10.1016/j.jaca.2024.100024","url":null,"abstract":"<div><div>The elementary theory of bivariate linear Diophantine equations over polynomial rings is used to construct causal lifting factorizations (elementary matrix decompositions) for causal two-channel FIR perfect reconstruction transfer matrices and wavelet transforms. The Diophantine approach generates causal factorizations satisfying certain polynomial degree-reducing inequalities, enabling a new factorization strategy called the <em>Causal Complementation Algorithm</em>. This provides a causal (i.e., polynomial, hence <em>realizable</em>) alternative to the noncausal lifting scheme developed by Daubechies and Sweldens using the Extended Euclidean Algorithm for Laurent polynomials. The new approach replaces the Euclidean Algorithm with Gaussian elimination employing a slight generalization of polynomial division that ensures existence and uniqueness of quotients whose remainders satisfy user-specified divisibility constraints. The Causal Complementation Algorithm is shown to be more general than the causal version of the Euclidean Algorithm approach by generating additional causal lifting factorizations beyond those obtainable using the polynomial Euclidean Algorithm.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"12 ","pages":"Article 100024"},"PeriodicalIF":0.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New condensation methods with applications to the computation of Brauer character tables","authors":"Klaus Lux ,&nbsp;A.J.E. Ryba","doi":"10.1016/j.jaca.2024.100023","DOIUrl":"10.1016/j.jaca.2024.100023","url":null,"abstract":"<div><div>Condensation is a technique that can often predict a Brauer character table of a finite group with a very high degree of confidence, but without a proof of correctness. In this paper we describe a strategy that can give such a proof. We introduce and apply two novel condensation methods: virtual tensor condensation and the condensation of bilinear forms. We illustrate our strategy and new techniques with examples taken from our computation of the 5-modular Brauer character table of the sporadic simple Lyons group.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"12 ","pages":"Article 100023"},"PeriodicalIF":0.0,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142419705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}