Finite Fields and Their Applications最新文献

{"title":"Some permutation pentanomials over finite fields of even characteristic","authors":"Farhana Kousar ,&nbsp;Maosheng Xiong","doi":"10.1016/j.ffa.2025.102742","DOIUrl":"10.1016/j.ffa.2025.102742","url":null,"abstract":"<div><div>In a recent paper <span><span>[30]</span></span> Zhang et al. constructed 17 families of permutation pentanomials of the form <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>. In this paper for 14 of these 17 families we provide a simple explanation as to why they are permutations. We also extend these 14 families into three general classes of permutation pentanomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102742"},"PeriodicalIF":1.2,"publicationDate":"2025-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362888","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":"Finite fields whose members are the sum of a potent and a 4-potent","authors":"Stephen D. Cohen ,&nbsp;Peter V. Danchev ,&nbsp;Tomás Oliveira e Silva","doi":"10.1016/j.ffa.2025.102739","DOIUrl":"10.1016/j.ffa.2025.102739","url":null,"abstract":"<div><div>We classify those finite fields <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> whose members are the sum of an <em>n</em>-potent element with <span><math><mi>n</mi><mo>&gt;</mo><mn>1</mn></math></span> and a 4-potent element. It is shown that there are precisely ten non-trivial pairs <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> for which this is the case. This continues a recent publication by Abyzov et al. (2024) <span><span>[1]</span></span> in which the tripotent version was examined in-depth, inasmuch as it extends recent results in this seam of research established by Abyzov and Tapkin (2024) <span><span>[4]</span></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102739"},"PeriodicalIF":1.2,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362887","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 sum-free functions","authors":"Alyssa Ebeling ,&nbsp;Xiang-dong Hou ,&nbsp;Ashley Rydell ,&nbsp;Shujun Zhao","doi":"10.1016/j.ffa.2025.102744","DOIUrl":"10.1016/j.ffa.2025.102744","url":null,"abstract":"<div><div>A function from <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> to <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> is said to be <em>kth order sum-free</em> if the sum of its values over each <em>k</em>-dimensional <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-affine subspace of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> is nonzero. This notion was recently introduced by C. Carlet as, among other things, a generalization of APN functions. At the center of this new topic is a conjecture about the sum-freedom of the multiplicative inverse function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mtext>inv</mtext></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> (with <span><math><msup><mrow><mn>0</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> defined to be 0). It is known that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mtext>inv</mtext></mrow></msub></math></span> is 2nd order (equivalently, <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></math></span>th order) sum-free if and only if <em>n</em> is odd, and it is conjectured that for <span><math><mn>3</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>3</mn></math></span>, <span><math><msub><mrow><mi>f</mi></mrow><mrow><mtext>inv</mtext></mrow></msub></math></span> is never <em>k</em>th order sum-free. The conjecture has been confirmed for even <em>n</em> but remains open for odd <em>n</em>. In the present paper, we show that the conjecture holds under each of the following conditions: (1) <span><math><mi>n</mi><mo>=</mo><mn>13</mn></math></span>; (2) <span><math><mn>3</mn><mo>|</mo><mi>n</mi></math></span>; (3) <span><math><mn>5</mn><mo>|</mo><mi>n</mi></math></span>; (4) the smallest prime divisor <em>l</em> of <em>n</em> satisfies <span><math><mo>(</mo><mi>l</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>l</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>≤</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span>. We also determine the “right” <em>q</em>-ary generalization of the binary multiplicative inverse function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mtext>inv</mtext></mrow></msub></math></span> in the context of sum-freedom. This <em>q</em>-ary generalization not only maintains most results for its binary version, but also exhibits some extraordinary phenomena that are not observed in the binary case.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102744"},"PeriodicalIF":1.2,"publicationDate":"2025-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362889","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":"Further results on permutation pentanomials over Fq3 in characteristic two","authors":"Tongliang Zhang ,&nbsp;Lijing Zheng ,&nbsp;Hengtai Wang ,&nbsp;Jie Peng ,&nbsp;Yanjun Li","doi":"10.1016/j.ffa.2025.102743","DOIUrl":"10.1016/j.ffa.2025.102743","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 a recent paper <span><span>[34]</span></span>, Zhang and Zheng investigated 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><mn>3</mn></mrow></msup></mrow></msub><mspace></mspace><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></math></span> with a certain linearized polynomial <span><math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. They applied the multivariate method and specific techniques to analyze the number of solutions of certain equations, and proposed an open problem: the permutation property of some pentanomials of this form remains unproven. In this paper, inspired by the idea of <span><span>[12]</span></span>, we further characterize the permutation property of such pentanomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub><mspace></mspace><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></math></span>. The techniques presented in this paper will be useful for investigating more new classes of permutation polynomials.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102743"},"PeriodicalIF":1.2,"publicationDate":"2025-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362890","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":"Binary polycyclic codes associated with x2η+1+x2η+1: Hamming distance, duality, reversibility and LCD properties","authors":"Sujata Bansal,&nbsp;Pramod Kumar Kewat","doi":"10.1016/j.ffa.2025.102741","DOIUrl":"10.1016/j.ffa.2025.102741","url":null,"abstract":"<div><div>This work explores binary polycyclic codes associated with the polynomial <span><math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>η</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>η</mi></mrow></msup></mrow></msup><mo>+</mo><mn>1</mn></math></span>, which is the <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>η</mi></mrow></msup></math></span>-th power of <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>x</mi><mo>+</mo><mn>1</mn></math></span> for every integer <span><math><mi>η</mi><mo>≥</mo><mn>1</mn></math></span>. We provide an in-depth structural analysis of these codes and compute the exact Hamming distance of each of these binary polycyclic codes. Furthermore, we determine the parity-check matrices and examine the Euclidean duals and annihilator duals for these polycyclic codes. Our analysis reveals that these codes are reversible and, in certain cases, are Linear Complementary Dual (LCD) codes. This discovery highlights the potential of these codes in practical applications such as communication systems, data storage, consumer electronics, and cryptography. We also propose a conjecture that suggests all such polycyclic codes can be LCD.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102741"},"PeriodicalIF":1.2,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321166","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":"Irreducible factorizations of polynomials xpk+1−bx+b over a finite field","authors":"Xue Jia ,&nbsp;Fengwei Li ,&nbsp;Huan Sun ,&nbsp;Qin Yue","doi":"10.1016/j.ffa.2025.102740","DOIUrl":"10.1016/j.ffa.2025.102740","url":null,"abstract":"<div><div>In this paper, we investigate polynomials of the form <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>b</mi></math></span>, where <span><math><mn>0</mn><mo>≠</mo><mi>b</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, <em>p</em> is a prime, and <em>k</em> divides <em>n</em>. By introducing a new approach based on the projective general linear group, we show that the number of zeros of <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> in <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> belongs to <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>}</mo></math></span>, and provide explicit criteria on <em>b</em> for each case. We also count the number of such polynomials corresponding to each possible number of zeros. Moreover, for the cases where <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> has at least one zero, we determine its complete irreducible factorization over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102740"},"PeriodicalIF":1.2,"publicationDate":"2025-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321165","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":"An unusual family of supersingular curves of genus five in characteristic two","authors":"Dušan Dragutinović","doi":"10.1016/j.ffa.2025.102736","DOIUrl":"10.1016/j.ffa.2025.102736","url":null,"abstract":"<div><div>We construct a family of smooth supersingular curves of genus 5 in characteristic 2 with several notable features: its dimension matches the expected dimension of any component of the supersingular locus in genus 5, its members are non-hyperelliptic curves with non-trivial automorphism groups, and each curve in the family admits a double cover structure over both an elliptic curve and a genus-2 curve. We also provide an explicit parametrization of this family.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102736"},"PeriodicalIF":1.2,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267945","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 a class of complete permutation quadrinomials","authors":"Chin Hei Chan ,&nbsp;Zhiguo Ding ,&nbsp;Nian Li ,&nbsp;Xi Xie ,&nbsp;Maosheng Xiong ,&nbsp;Michael E. Zieve","doi":"10.1016/j.ffa.2025.102734","DOIUrl":"10.1016/j.ffa.2025.102734","url":null,"abstract":"<div><div>Let <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn><mi>q</mi></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>+</mo><mi>d</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, where <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> is the finite field of order <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span> for some positive integer <em>m</em>. Tu et al. (Finite Fields Appl. 68: 1-20, 2020) proposed a sufficient condition under which <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is a complete permutation on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>. In this paper, we show that this sufficient condition is also necessary, and when <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is a complete permutation, then <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>x</mi></math></span> are simultaneously linear equivalent to <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mover><mrow><mi>x</mi></mrow><mo>‾</mo></mover></math></span> and <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mover><mrow><mi>x</mi></mrow><mo>‾</mo></mover><mo>+</mo><mi>γ</mi><mi>x</mi></math></span> for some <span><math><mi>γ</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> satisfying <span><math><mrow><mi>ord</mi></mrow><mo>(</mo><msup><mrow><mi>γ</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo><mo>=</mo><mn>3</mn></math></span>. This result leads to a complete characterization of the complete permutation quadrinomials of the above form <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102734"},"PeriodicalIF":1.2,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267944","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 multiplicity-one theorem for the superspeciality of curves of genus two","authors":"Shushi Harashita,&nbsp;Yuya Yamamoto","doi":"10.1016/j.ffa.2025.102738","DOIUrl":"10.1016/j.ffa.2025.102738","url":null,"abstract":"<div><div>Igusa proved in 1958 that the polynomial determining the supersingularity of elliptic curves in Legendre form is separable. In this paper, we get an analogous result for curves of genus 2 in Rosenhain form. More precisely we show that the ideal determining the superspeciality of the curve has multiplicity one at every superspecial point. Igusa used a Picard-Fucks differential operator annihilating a Gauß hypergeometric series. We shall use the Lauricella system (of type D) of hypergeometric differential equations in three variables.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102738"},"PeriodicalIF":1.2,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267943","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 small densities defined without pseudorandomness","authors":"Thomas Karam","doi":"10.1016/j.ffa.2025.102735","DOIUrl":"10.1016/j.ffa.2025.102735","url":null,"abstract":"<div><div>We identify a new sufficient condition on linear forms <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>:</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>→</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> which guarantees that every subset of <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> on which none of <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> has full image has a density which tends to 0 with <em>k</em>. The condition is much weaker than the condition usually used to guarantee that <span><math><mo>(</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> takes each value of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math></span> with probability close to <span><math><msup><mrow><mi>p</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msup></math></span> when <em>x</em> is chosen uniformly at random in the Boolean cube <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span>. The density is at most quasipolynomially small in <em>k</em>, a bound that is necessarily close to sharp.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102735"},"PeriodicalIF":1.2,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267946","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}