Finite Fields and Their Applications最新文献

{"title":"On quadratic character sums over quartics","authors":"Bogdan Nica","doi":"10.1016/j.ffa.2025.102755","DOIUrl":"10.1016/j.ffa.2025.102755","url":null,"abstract":"<div><div>We obtain transformation formulas for quadratic character sums with quartic and cubic polynomial arguments.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102755"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466208","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":"Mean value theorems for short rational exponential sums","authors":"Doowon Koh ,&nbsp;Igor E. Shparlinski","doi":"10.1016/j.ffa.2025.102719","DOIUrl":"10.1016/j.ffa.2025.102719","url":null,"abstract":"<div><div>We obtain finite field analogues of a series of recent results on various mean value theorems for Weyl sums. Instead of the Vinogradov Mean Value Theorem, our results rest on the classical argument of Mordell, combined with several other ideas.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102719"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145020639","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":"2026-02-01","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":"Continued fractions and indefinite binary quadratic forms over Fq[t]","authors":"Jorge Morales","doi":"10.1016/j.ffa.2025.102747","DOIUrl":"10.1016/j.ffa.2025.102747","url":null,"abstract":"<div><div>The relation between cycles of indefinite binary quadratic forms over <span><math><mi>Z</mi></math></span> and continued fractions is classical and well-known. We describe a similar relation for binary quadratic forms over the polynomial ring <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>t</mi><mo>]</mo></math></span>, where <em>q</em> is a power of an odd prime. In this context, the cycles of the classical theory are replaced by orbits of the metacyclic group <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>⋊</mo><mi>Z</mi></math></span> acting on the set of reduced forms of a given discriminant, where each orbit corresponds to a proper equivalence class.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102747"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416452","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":"Cardinality-consistent flag codes with larger cardinality","authors":"Junfeng Jia,&nbsp;Yanxun Chang","doi":"10.1016/j.ffa.2025.102750","DOIUrl":"10.1016/j.ffa.2025.102750","url":null,"abstract":"<div><div>Flag codes, as a generalization of subspace codes, can transmit more information since the subspace channel is used many times. In this paper, we construct optimum distance flag codes of the (generalized) full admissible type <span><math><mi>t</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> on <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> with cardinality <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msup><mrow><mi>q</mi></mrow><mrow><mi>i</mi><mi>k</mi><mo>+</mo><mi>h</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span>, where <span><math><mi>n</mi><mo>=</mo><mi>s</mi><mi>k</mi><mo>+</mo><mi>h</mi></math></span> with <span><math><mi>s</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mn>0</mn><mo>≤</mo><mi>h</mi><mo>&lt;</mo><mi>k</mi></math></span>. Let <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>f</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>,</mo><mi>t</mi><mo>)</mo></math></span> denote the maximum cardinality of such codes. We provide a lower bound for this quantity. We further present a systematic construction of cardinality-consistent flag codes with larger cardinality for general flag distances. By the composition of subspace polynomials, we construct cardinality-consistent cyclic flag codes on <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 larger cardinality than those presented in the literature.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102750"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416454","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":"2026-02-01","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":"The dual codes of two families of BCH codes","authors":"Haojie Xu ,&nbsp;Xia Wu ,&nbsp;Wei Lu ,&nbsp;Xiwang Cao","doi":"10.1016/j.ffa.2025.102721","DOIUrl":"10.1016/j.ffa.2025.102721","url":null,"abstract":"<div><div>In this paper, we present an infinite family of MDS codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>s</mi></mrow></msup></mrow></msub></math></span> and two infinite families of almost MDS codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup></mrow></msub></math></span> for any prime <em>p</em>, by investigating the parameters of the dual codes of two families of BCH codes. Notably, these almost MDS codes include two infinite families of near MDS codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>s</mi></mrow></msup></mrow></msub></math></span>, resolving a conjecture posed by Geng et al. in 2022. Furthermore, we demonstrate that both of these almost MDS codes and their dual codes hold infinite families of 3-designs over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup></mrow></msub></math></span> for any prime <em>p</em>. Additionally, we study the subfield subcodes of these families of MDS and near MDS codes, and provide several binary, ternary, and quaternary codes with best known parameters.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102721"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097424","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":"2026-02-01","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":"Cryptanalysis of some algebraic variants of the RSA cryptosystem","authors":"Mohammed Rahmani ,&nbsp;Abderrahmane Nitaj ,&nbsp;Mhammed Ziane","doi":"10.1016/j.ffa.2025.102757","DOIUrl":"10.1016/j.ffa.2025.102757","url":null,"abstract":"<div><div>Let <span><math><mi>N</mi><mo>=</mo><mi>p</mi><mi>q</mi></math></span> be an RSA modulus, and <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> be an integer. Two recently algebraic variants of the RSA cryptosystem use a public exponent <em>e</em> for encryption, and a private exponent <em>d</em> for decryption with <span><math><mi>e</mi><mi>d</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>=</mo><mrow><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>. In this paper, we propose an attack on the two variants using Coppersmith's method and lattice basis reduction. Our attack breaks the systems when <em>d</em> is less than an explicit bound that depends only on <em>n</em> and <em>N</em>. We analyze the security of the RSA variants characterized by the equation <span><math><mi>e</mi><mi>d</mi><mo>−</mo><mi>k</mi><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Specifically, we propose a novel attack utilizing lattice-based methods and Coppersmith's technique, when the prime numbers <em>p</em> and <em>q</em> share an amount of their least significant bits. This enables the efficient recovery of the primes <em>p</em> and <em>q</em> in polynomial time.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102757"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466204","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":"Brocard-Ramanujan problem for polynomials over finite fields","authors":"Wataru Takeda","doi":"10.1016/j.ffa.2025.102731","DOIUrl":"10.1016/j.ffa.2025.102731","url":null,"abstract":"<div><div>The Brocard-Ramanujan problem is an unsolved number theory problem to find integer solutions <span><math><mo>(</mo><mi>x</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> to <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><mi>n</mi><mo>!</mo></math></span>. In this paper, we consider this problem over polynomial rings <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>T</mi><mo>]</mo></math></span>, where <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is a finite field with <em>q</em> elements. We find all solutions to the equation <span><math><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><msub><mrow><mi>Π</mi></mrow><mrow><mi>C</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mi>C</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denotes the Carlitz factorial. More precisely, we characterize all solutions and prove that there are infinitely many solutions if and only if <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is an extension of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. This characterization is achieved without using the Mason-Stothers theorem, analogous to the abc conjecture for integers.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102731"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145221461","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}