Pub Date : 2025-03-01DOI: 10.1016/j.jaca.2025.100033
Ralf Fröberg
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.
{"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}
Pub Date : 2025-03-01DOI: 10.1016/j.jaca.2025.100034
Colin Ramsay
Lists of equivalence classes of words under rotation or rotation plus reversal (i.e., necklaces and bracelets) have many uses, and efficient algorithms for generating these lists exist. In combinatorial group theory elements of a group are typically written as words in the generators and their inverses, and necklaces and bracelets correspond to conjugacy classes and relators respectively. We present algorithms to generate lists of freely and cyclically reduced necklaces and bracelets in free groups. Experimental evidence suggests that these algorithms are CAT – that is, they run in constant amortized time.
{"title":"Listing words in free groups","authors":"Colin Ramsay","doi":"10.1016/j.jaca.2025.100034","DOIUrl":"10.1016/j.jaca.2025.100034","url":null,"abstract":"<div><div>Lists of equivalence classes of words under rotation or rotation plus reversal (i.e., necklaces and bracelets) have many uses, and efficient algorithms for generating these lists exist. In combinatorial group theory elements of a group are typically written as words in the generators and their inverses, and necklaces and bracelets correspond to conjugacy classes and relators respectively. We present algorithms to generate lists of freely and cyclically reduced necklaces and bracelets in free groups. Experimental evidence suggests that these algorithms are CAT – that is, they run in constant amortized time.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"13 ","pages":"Article 100034"},"PeriodicalIF":0.0,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144190158","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}
Pub Date : 2025-03-01DOI: 10.1016/j.jaca.2025.100030
J. William Hoffman , Haohao Wang
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 μ-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.
{"title":"Computational and theoretical aspects of rational parametrization of generalized tubular surfaces","authors":"J. William Hoffman , 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}
Pub Date : 2025-03-01DOI: 10.1016/j.jaca.2025.100035
Kirpa Garg , Sartaj Ul Hasan , Pantelimon Stănică
The notion of c-differential uniformity has received a lot of attention since its proposal [5], and recently a characterization of perfect c-nonlinear functions in terms of difference sets in some quasigroups was obtained in [1]. Moreover, in a very recent manuscript by Pal and Stănică [19], an intriguing connection was discovered showing that in fact, the boomerang uniformity for an odd APN function (odd characteristic) equals its c-differential uniformity when , if the function is a permutation, otherwise it is the maximum of the -DDT entries disregarding the first row/column. The construction of functions, especially permutations, with low c-differential uniformity is an interesting and difficult mathematical problem in this area, and recent work has focused heavily in this direction. We provide a few classes of permutation polynomials with low c-differential uniformity. The used technique involves handling various Weil sums, as well as analyzing some equations in finite fields, and we believe these can be of independent interest, from a mathematical perspective.
{"title":"Differential uniformity properties of some classes of permutation polynomials","authors":"Kirpa Garg , Sartaj Ul Hasan , Pantelimon Stănică","doi":"10.1016/j.jaca.2025.100035","DOIUrl":"10.1016/j.jaca.2025.100035","url":null,"abstract":"<div><div>The notion of <em>c</em>-differential uniformity has received a lot of attention since its proposal <span><span>[5]</span></span>, and recently a characterization of perfect <em>c</em>-nonlinear functions in terms of difference sets in some quasigroups was obtained in <span><span>[1]</span></span>. Moreover, in a very recent manuscript by Pal and Stănică <span><span>[19]</span></span>, an intriguing connection was discovered showing that in fact, the boomerang uniformity for an odd APN function (odd characteristic) equals its <em>c</em>-differential uniformity when <span><math><mi>c</mi><mo>=</mo><mo>−</mo><mn>1</mn></math></span>, if the function is a permutation, otherwise it is the maximum of the <span><math><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-DDT entries disregarding the first row/column. The construction of functions, especially permutations, with low <em>c</em>-differential uniformity is an interesting and difficult mathematical problem in this area, and recent work has focused heavily in this direction. We provide a few classes of permutation polynomials with low <em>c</em>-differential uniformity. The used technique involves handling various Weil sums, as well as analyzing some equations in finite fields, and we believe these can be of independent interest, from a mathematical perspective.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"13 ","pages":"Article 100035"},"PeriodicalIF":0.0,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144240750","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}
Pub Date : 2025-03-01DOI: 10.1016/j.jaca.2025.100031
Thomas Breuer , Michael Joswig , Gunter Malle
The concept of S-characters of finite groups was introduced by Zhmud' as a generalisation of transitive permutation characters. Any non-trivial S-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 S-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 OSCAR.
{"title":"Zeros of S-characters","authors":"Thomas Breuer , Michael Joswig , 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}
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