{"title":"On the sumsets of units and exceptional units in residue class rings","authors":"Siao Hong","doi":"10.1016/j.ffa.2024.102566","DOIUrl":"10.1016/j.ffa.2024.102566","url":null,"abstract":"<div><div>Let <span><math><mi>n</mi><mo>≥</mo><mn>1</mn><mo>,</mo><mi>e</mi><mo>≥</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <em>c</em> be integers such that <span><math><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>k</mi></math></span>. An integer <em>u</em> is called a unit in the ring <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of residue classes modulo <em>n</em> if <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Let <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> be the multiplicative group of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. A unit <em>u</em> is called an exceptional unit in the ring <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> if <span><math><mn>1</mn><mo>−</mo><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. We write <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo><mo>⁎</mo></mrow></msubsup></math></span> for the set of all exceptional units of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We denote by <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>e</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> the number of representations of the element <span><math><mi>c</mi><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> as the sum of <em>e</em>-th powers of <em>t</em> units and <em>e</em>-th powers of <span><math><mi>k</mi><mo>−</mo><mi>t</mi></math></span> exceptional units in the ring <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. When <span><math><mi>t</mi><mo>=</mo><mi>k</mi></math></span>, Brauer determined the number <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>c</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> which answers a question of Rademacher. Mollahajiaghaei gave a formula for <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>c</mi><mo>,</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. When <span><math><mi>t</mi><mo>=</mo><mn>0</mn></math></span>, Sander presented a formula for <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>c</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, and later on Yang and Zhao got an exact formula for <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>k</mi><mo>,</mo><mi>c</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102566"},"PeriodicalIF":1.2,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143135116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The complexity of elliptic normal bases","authors":"Daniel Panario , Mohamadou Sall , Qiang Wang","doi":"10.1016/j.ffa.2024.102570","DOIUrl":"10.1016/j.ffa.2024.102570","url":null,"abstract":"<div><div>We study the complexity (that is, the weight of the multiplication table) of the elliptic normal bases introduced by Couveignes and Lercier. We give an upper bound on the complexity of these elliptic normal bases, and we analyze the weight of some specific vectors related to the multiplication table of those bases. This analysis leads us to some perspectives on the search for low complexity normal bases from elliptic periods.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102570"},"PeriodicalIF":1.2,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143135117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"More classes of permutation pentanomials over finite fields with even characteristic","authors":"Tongliang Zhang , Lijing Zheng","doi":"10.1016/j.ffa.2024.102567","DOIUrl":"10.1016/j.ffa.2024.102567","url":null,"abstract":"<div><div>Let <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>. In this paper, we propose several classes of permutation pentanomials of the form <span><math><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup><mo>+</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>+</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mspace></mspace><mo>(</mo><mn>2</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>4</mn><mo>)</mo></math></span> from some certain linearized polynomial <span><math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> by using multivariate method and some techniques to determine the solutions of some equations. Furthermore, two classes of permutation pentanomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> for <em>n</em> satisfying <span><math><mn>3</mn><mo>|</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span> are also constructed based on some bijections over the unit circle <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>τ</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> with order <span><math><mi>τ</mi><mo>=</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102567"},"PeriodicalIF":1.2,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A correct justification for the CHMT algorithm for solving underdetermined multivariate systems","authors":"Daniel Smith-Tone , Cristina Tone","doi":"10.1016/j.ffa.2024.102547","DOIUrl":"10.1016/j.ffa.2024.102547","url":null,"abstract":"<div><div>Cheng et al. (2014) <span><span>[6]</span></span> introduced a substantial improvement to the Miura-Hashimoto-Takagi algorithm for solving sufficiently underdetermined systems of multivariate polynomial equations. This improvement claimed to make the algorithm polynomial time for instances satisfying <span><math><mi>n</mi><mo>≥</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>, where <em>m</em> is the number of equations and <em>n</em> is the number of variables. While experimentally, the algorithm seems to work, we have uncovered a subtle error in the proof of time complexity for the algorithm. Due to the fact that there have been multiple proposals for algorithms based on this and related algorithms, as well as the recent submission to NIST's call for additional post-quantum digital signatures of a more modern “provably secure” version of the famous UOV digital signature algorithm based on the foundational structure of this algorithm, our observation may highlight a concerning theoretical deficiency in this area of research.</div><div>In this work, we provide a tight justification for the polynomial time complexity of the algorithm (with a very minor tweak), thereby justifying the complexity of enhancements based upon it as well. At the heart of this justification is a precise calculation of the probability of recovering a maximal depth path in polynomially many steps within a possibly exponentially large search tree. While this algorithmic problem is generic, we find that the parameters relevant for the application to the above algorithm are extremal and poorly studied. Thus, our analysis serves to clarify the boundary behavior of such search algorithms with respect to complexity classes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102547"},"PeriodicalIF":1.2,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the square code of group codes","authors":"Alejandro Piñera Nicolás , Ignacio Fernández Rúa , Adriana Suárez Corona","doi":"10.1016/j.ffa.2024.102548","DOIUrl":"10.1016/j.ffa.2024.102548","url":null,"abstract":"<div><div>Error correcting codes have recently gained more attention due to their applications in quantum resistant cryptography. Their suitability depends on their indistinguishability from random codes. In that sense, the study of the square code of a particular code provides a tool for distinguishing random codes from not random ones.</div><div>With this motivation, the square codes of some semisimple bilateral group codes, as abelian and dihedral ones, are studied in this paper. For this purpose, bilateral group codes are described as evaluation codes by means of the absolutely irreducible characters of the group. Finally, some results on self-duality and self-orthogonality are recovered under this alternative point of view.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102548"},"PeriodicalIF":1.2,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The R-transform as power map and its generalizations to higher degree","authors":"Alp Bassa , Ricardo Menares","doi":"10.1016/j.ffa.2024.102546","DOIUrl":"10.1016/j.ffa.2024.102546","url":null,"abstract":"<div><div>We give iterative constructions for irreducible polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of degree <span><math><mi>n</mi><mo>⋅</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> for all <span><math><mi>r</mi><mo>≥</mo><mn>0</mn></math></span>, starting from irreducible polynomials of degree <em>n</em>. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup></math></span>. The <em>R</em>-transform introduced by Cohen is recovered as a particular case corresponding to <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, hence we obtain a generalization of Cohen's <em>R</em>-transform (<span><math><mi>t</mi><mo>=</mo><mn>2</mn></math></span>) to arbitrary degrees <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span>. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> we recover and generalize a recursive construction of Panario, Reis and Wang.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102546"},"PeriodicalIF":1.2,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Webs and squabs of conics over finite fields","authors":"Nour Alnajjarine , Michel Lavrauw","doi":"10.1016/j.ffa.2024.102544","DOIUrl":"10.1016/j.ffa.2024.102544","url":null,"abstract":"<div><div>This paper is a contribution towards a solution for the longstanding open problem of classifying linear systems of conics over finite fields initiated by L. E. Dickson in 1908, through his study of the projective equivalence classes of pencils of conics in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, for <em>q</em> odd. In this paper a set of complete invariants is determined for the projective equivalence classes of webs and of squabs of conics in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, both for <em>q</em> odd and even. Our approach is mainly geometric, and involves a comprehensive study of the geometric and combinatorial properties of the Veronese surface in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>. The main contribution is the determination of the distribution of the different types of hyperplanes incident with the <em>K</em>-orbit representatives of points and lines of <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, where <span><math><mi>K</mi><mo>≅</mo><mrow><mi>PGL</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, is the subgroup of <span><math><mrow><mi>PGL</mi></mrow><mo>(</mo><mn>6</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> stabilizing the Veronese surface.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102544"},"PeriodicalIF":1.2,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some q-ary constacyclic BCH codes with length qm+12","authors":"Jin Li , Huilian Zhu , Shixin Zhu","doi":"10.1016/j.ffa.2024.102545","DOIUrl":"10.1016/j.ffa.2024.102545","url":null,"abstract":"<div><div>In this paper, we study some <em>q</em>-ary <em>λ</em>-constacyclic BCH codes of length <span><math><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> with some large designed distances for <span><math><mrow><mi>ord</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span> and <span><math><mn>2</mn><mo>+</mo><mo>⌈</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mi>q</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>⌉</mo><mo>≤</mo><mrow><mi>ord</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>q</mi><mo>−</mo><mn>1</mn></math></span> respectively, where <em>q</em> is an odd prime power and <span><math><mrow><mi>ord</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>|</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. The dimensions and the lower bounds on the minimum distances of these codes are given by using recurrence relations and the introduced definitions of sequences. The code examples presented in this paper indicate that these codes have good parameters in general.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102545"},"PeriodicalIF":1.2,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142707366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Repeated-root constacyclic codes of length kslmpn over finite fields","authors":"Qi Zhang , Weiqiong Wang , Shuyu Luo , Yue Li","doi":"10.1016/j.ffa.2024.102542","DOIUrl":"10.1016/j.ffa.2024.102542","url":null,"abstract":"<div><div>For different odd primes <em>k</em>, <em>l</em>, <em>p</em>, and positive integers <em>s</em>, <em>m</em>, <em>n</em>, the polynomial <span><math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>l</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msup><mo>−</mo><mi>λ</mi></math></span> in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow></math></span> is explicitly factorized, where <em>p</em> is the characteristic of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, <span><math><mi>λ</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. All repeated-root constacyclic codes and their dual codes of length <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>l</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> are characterized. In addition, the characterization and enumeration of all linear complementary dual (LCD) cyclic and negacyclic codes of length <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>l</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> are obtained.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102542"},"PeriodicalIF":1.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Complete description of measures corresponding to Abelian varieties over finite fields","authors":"Nikolai S. Nadirashvili , Michael A. Tsfasman","doi":"10.1016/j.ffa.2024.102543","DOIUrl":"10.1016/j.ffa.2024.102543","url":null,"abstract":"<div><div>We study probability measures corresponding to families of abelian varieties over a finite field. These measures play an important role in the Tsfasman–Vlăduţ theory of asymptotic zeta-functions defining completely the limit zeta-function of the family. J.-P. Serre, using results of R.M. Robinson on conjugate algebraic integers, described the possible set of measures than can correspond to families of abelian varieties over a finite field. The problem whether all such measures actually occur was left open. Moreover, Serre supposed that not all such measures correspond to abelian varieties (for example, the Lebesgue measure on a segment). Here we settle Serre's problem proving that Serre conditions are sufficient, and thus describe completely the set of measures corresponding to abelian varieties.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102543"},"PeriodicalIF":1.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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