Pub Date : 2025-10-13DOI: 10.1016/j.jaca.2025.100041
Alexei Lisitsa
Recent work by Shehper et al. (2024) [13] proposed that the well-known Akbulut–Kirby presentation is stably Andrews–Curtis (AC) equivalent to the trivial presentation, based on a reduction to a previously studied presentation P. In this paper, we present an independently obtained reduction of to P, using a fully automated theorem proving approach. Subsequent to our initial submission, it has emerged that the earlier theoretical claim about the stable AC-trivializability of P—on which the Shehper et al. result relied—was based on an incorrect theorem, and thus the status of remains unresolved. While this paper does not establish the stable AC-triviality of , it contributes a reproducible and verifiable case study in the use of automated reasoning to analyze deep problems in combinatorial group theory. We conclude by suggesting future directions for computational exploration of stable AC-transformations.
{"title":"Stable Andrews-Curtis trivialization of AK(3) revisited. A case study using automated deduction","authors":"Alexei Lisitsa","doi":"10.1016/j.jaca.2025.100041","DOIUrl":"10.1016/j.jaca.2025.100041","url":null,"abstract":"<div><div>Recent work by Shehper et al. (2024) <span><span>[13]</span></span> proposed that the well-known Akbulut–Kirby presentation <span><math><mrow><mi>AK</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></math></span> is stably Andrews–Curtis (AC) equivalent to the trivial presentation, based on a reduction to a previously studied presentation <em>P</em>. In this paper, we present an independently obtained reduction of <span><math><mrow><mi>AK</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></math></span> to <em>P</em>, using a fully automated theorem proving approach. Subsequent to our initial submission, it has emerged that the earlier theoretical claim about the stable AC-trivializability of <em>P</em>—on which the Shehper et al. result relied—was based on an incorrect theorem, and thus the status of <span><math><mrow><mi>AK</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></math></span> remains unresolved. While this paper does not establish the stable AC-triviality of <span><math><mrow><mi>AK</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></math></span>, it contributes a reproducible and verifiable case study in the use of automated reasoning to analyze deep problems in combinatorial group theory. We conclude by suggesting future directions for computational exploration of stable AC-transformations.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"16 ","pages":"Article 100041"},"PeriodicalIF":0.0,"publicationDate":"2025-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145327132","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 : 2025-09-01DOI: 10.1016/j.jaca.2025.100040
Kymani T.K. Armstrong-Williams , Edward Hirst , Blake Jackson , Kyu-Hwan Lee
Machine learning (ML) has emerged as a powerful tool in mathematical research in recent years. This paper applies ML techniques to the study of quivers—a type of directed multigraph with significant relevance in algebra, combinatorics, computer science, and mathematical physics. Specifically, we focus on the challenging problem of determining the mutation-acyclicity of a quiver on 4 vertices, a property that is pivotal since mutation-acyclicity is often a necessary condition for theorems involving path algebras and cluster algebras. Although this classification is known for quivers with at most 3 vertices, little is known about quivers on more than 3 vertices. We give a computer-assisted proof of a theorem to prove that mutation-acyclicity is decidable for quivers on 4 vertices with edge weight at most 2. By leveraging neural networks (NNs) and support vector machines (SVMs), we then accurately classify more general 4-vertex quivers as mutation-acyclic or non-mutation-acyclic. Our results demonstrate that ML models can efficiently detect mutation-acyclicity, providing a promising computational approach to this combinatorial problem, from which the trained SVM equation provides a starting point to guide future theoretical development.
{"title":"Machine learning mutation-acyclicity of quivers","authors":"Kymani T.K. Armstrong-Williams , Edward Hirst , Blake Jackson , Kyu-Hwan Lee","doi":"10.1016/j.jaca.2025.100040","DOIUrl":"10.1016/j.jaca.2025.100040","url":null,"abstract":"<div><div>Machine learning (ML) has emerged as a powerful tool in mathematical research in recent years. This paper applies ML techniques to the study of quivers—a type of directed multigraph with significant relevance in algebra, combinatorics, computer science, and mathematical physics. Specifically, we focus on the challenging problem of determining the mutation-acyclicity of a quiver on 4 vertices, a property that is pivotal since mutation-acyclicity is often a necessary condition for theorems involving path algebras and cluster algebras. Although this classification is known for quivers with at most 3 vertices, little is known about quivers on more than 3 vertices. We give a computer-assisted proof of a theorem to prove that mutation-acyclicity is decidable for quivers on 4 vertices with edge weight at most 2. By leveraging neural networks (NNs) and support vector machines (SVMs), we then accurately classify more general 4-vertex quivers as mutation-acyclic or non-mutation-acyclic. Our results demonstrate that ML models can efficiently detect mutation-acyclicity, providing a promising computational approach to this combinatorial problem, from which the trained SVM equation provides a starting point to guide future theoretical development.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"15 ","pages":"Article 100040"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145018853","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 : 2025-07-23DOI: 10.1016/j.jaca.2025.100039
Andreas-Stephan Elsenhans , Jürgen Klüners
Given a number field, it is an important question in algorithmic number theory to determine all its subfields. If the search is restricted to abelian subfields, we can try to determine them by using class field theory. For this, it is necessary to know the ramified primes. We show that the ramified primes of the subfield can be computed efficiently. Using this information, we give algorithms to determine all quadratic and cyclic cubic subfields of the initial field. The approach generalises to cyclic subfields of prime degree. In the case of quadratic subfields, our approach is much faster than other methods.
{"title":"Computing quadratic subfields of number fields","authors":"Andreas-Stephan Elsenhans , Jürgen Klüners","doi":"10.1016/j.jaca.2025.100039","DOIUrl":"10.1016/j.jaca.2025.100039","url":null,"abstract":"<div><div>Given a number field, it is an important question in algorithmic number theory to determine all its subfields. If the search is restricted to abelian subfields, we can try to determine them by using class field theory. For this, it is necessary to know the ramified primes. We show that the ramified primes of the subfield can be computed efficiently. Using this information, we give algorithms to determine all quadratic and cyclic cubic subfields of the initial field. The approach generalises to cyclic subfields of prime degree. In the case of quadratic subfields, our approach is much faster than other methods.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"15 ","pages":"Article 100039"},"PeriodicalIF":0.0,"publicationDate":"2025-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144714107","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 : 2025-06-23DOI: 10.1016/j.jaca.2025.100038
Matthew B. Day , Trevor Nakamura
We construct a 3-dimensional cell complex that is the 3-skeleton for an Eilenberg–MacLane classifying space for the symmetric group . Our complex starts with the presentation for with adjacent transpositions with squaring, commuting, and braid relations, and adds seven classes of 3-cells that fill in certain 2-spheres bounded by these relations. We use a rewriting system and a combinatorial method of K. Brown to prove the correctness of our construction. Our main application is a computation of the second cohomology of in certain twisted coefficient modules; we use this computation in a companion paper to study splitting of extensions related to braid groups. As another application, we give a concrete description of the third homology of with untwisted coefficients in .
{"title":"A 3-skeleton for a classifying space for the symmetric group","authors":"Matthew B. Day , Trevor Nakamura","doi":"10.1016/j.jaca.2025.100038","DOIUrl":"10.1016/j.jaca.2025.100038","url":null,"abstract":"<div><div>We construct a 3-dimensional cell complex that is the 3-skeleton for an Eilenberg–MacLane classifying space for the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Our complex starts with the presentation for <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> adjacent transpositions with squaring, commuting, and braid relations, and adds seven classes of 3-cells that fill in certain 2-spheres bounded by these relations. We use a rewriting system and a combinatorial method of K. Brown to prove the correctness of our construction. Our main application is a computation of the second cohomology of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in certain twisted coefficient modules; we use this computation in a companion paper to study splitting of extensions related to braid groups. As another application, we give a concrete description of the third homology of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with untwisted coefficients in <span><math><mi>Z</mi></math></span>.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"15 ","pages":"Article 100038"},"PeriodicalIF":0.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517258","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 : 2025-06-19DOI: 10.1016/j.jaca.2025.100037
Jorge Fariña-Asategui
An explicit algorithm is given for the computation of the Hausdorff dimension of the closure of a regular branch group in terms of an arbitrary branch structure. We implement this algorithm in GAP and apply it to a family of GGS-groups acting on the 4-adic tree.
{"title":"An algorithm to compute the Hausdorff dimension of regular branch groups","authors":"Jorge Fariña-Asategui","doi":"10.1016/j.jaca.2025.100037","DOIUrl":"10.1016/j.jaca.2025.100037","url":null,"abstract":"<div><div>An explicit algorithm is given for the computation of the Hausdorff dimension of the closure of a regular branch group in terms of an arbitrary branch structure. We implement this algorithm in GAP and apply it to a family of GGS-groups acting on the 4-adic tree.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"15 ","pages":"Article 100037"},"PeriodicalIF":0.0,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144337924","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.2025.100032
Joseph Tonien
Ramanujan discovered the following elegant identity involving cube roots:
The goal of this paper is to derive nested fifth root radical identities in the form of 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.
{"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}
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}
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