Research in Number Theory最新文献

Transcendental Brauer-Manin obstructions on singular K3 surfaces. 奇异K3曲面上的先验Brauer-Manin障碍。

IF 0.6

Research in Number Theory Pub Date : 2025-01-01 Epub Date: 2024-12-18 DOI: 10.1007/s40993-024-00580-z

Mohamed Alaa Tawfik, Rachel Newton

引用次数: 0

Distribution of Andrews’ singular overpartitions $${overline{C}}_{p,1}(n)$$ C ¯ 安德鲁斯奇异超分区的分布 $${overline{C}}_{p,1}(n)$$ C ¯

IF 0.8

Research in Number Theory Pub Date : 2024-01-04 DOI: 10.1007/s40993-023-00496-0

Chiranjit Ray

引用次数: 0

Asymptotics of commuting ℓ -tuples in symmetric groups and log-concavity. 对称群中交换 ℓ -图元的渐近性和对数凹性

IF 0.6

Research in Number Theory Pub Date : 2024-01-01 Epub Date: 2024-10-03 DOI: 10.1007/s40993-024-00562-1

Kathrin Bringmann, Johann Franke, Bernhard Heim

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Asymptotics of commuting <ns0:math><ns0:mi>ℓ</ns0:mi></ns0:math> -tuples in symmetric groups and log-concavity.","authors":"Kathrin Bringmann, Johann Franke, Bernhard Heim","doi":"10.1007/s40993-024-00562-1","DOIUrl":"10.1007/s40993-024-00562-1","url":null,"abstract":"Denote by <math> <mrow><msub><mi>N</mi> <mi>ℓ</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> the number of <math><mi>ℓ</mi></math> -tuples of elements in the symmetric group <math><msub><mi>S</mi> <mi>n</mi></msub> </math> with commuting components, normalized by the order of <math><msub><mi>S</mi> <mi>n</mi></msub> </math> . In this paper, we prove asymptotic formulas for <math> <mrow><msub><mi>N</mi> <mi>ℓ</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> . In addition, general criteria for log-concavity are shown, which can be applied to <math> <mrow><msub><mi>N</mi> <mi>ℓ</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> among other examples. Moreover, we obtain a Bessenrodt-Ono type theorem which gives an inequality of the form <math><mrow><mi>c</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> <mi>c</mi> <mo>(</mo> <mi>b</mi> <mo>)</mo> <mo>></mo> <mi>c</mi> <mo>(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>)</mo></mrow> </math> for certain families of sequences c(n).","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"10 4","pages":"83"},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11449981/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142381962","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}

引用次数: 0

Log concavity for unimodal sequences. 单模态序列的对数凹性。

IF 0.8

Research in Number Theory Pub Date : 2024-01-01 Epub Date: 2023-12-19 DOI: 10.1007/s40993-023-00490-6

Walter Bridges, Kathrin Bringmann

引用次数: 0

On Pillai's Problem involving Lucas sequences of the second kind. 论涉及第二类卢卡斯序列的皮莱问题

IF 0.6

Research in Number Theory Pub Date : 2024-01-01 Epub Date: 2024-05-13 DOI: 10.1007/s40993-024-00534-5

Sebastian Heintze, Volker Ziegler

引用次数: 0

Odd degree isolated points on $$X_1(N)$$ with rational j-invariant 在$$X_1(N)$$上的奇数孤立点具有理性 j 不变性

IF 0.8

Research in Number Theory Pub Date : 2023-12-17 DOI: 10.1007/s40993-023-00488-0

Abbey Bourdon, David R. Gill, Jeremy Rouse, Lori D. Watson

引用次数: 0

Error approximation for backwards and simple continued fractions 倒数和简单续分数的误差近似值

IF 0.8

Research in Number Theory Pub Date : 2023-11-22 DOI: 10.1007/s40993-023-00481-7

Cameron Bjorklund, Matthew Litman

引用次数: 0

Transcendence criterion with $$(beta ,{mathcal {A}})$$-representations in some quadratic integer bases 二次整数基中$$(beta ,{mathcal {A}})$$ -表示的超越判据

Research in Number Theory Pub Date : 2023-11-10 DOI: 10.1007/s40993-023-00486-2

Maryam Elaoud, Mohamed Hbaib

引用次数: 0

Fast norm computation in smooth-degree Abelian number fields 光滑阿贝尔数域的快速范数计算

Research in Number Theory Pub Date : 2023-11-10 DOI: 10.1007/s40993-022-00402-0

Daniel J. Bernstein

{"title":"Fast norm computation in smooth-degree Abelian number fields","authors":"Daniel J. Bernstein","doi":"10.1007/s40993-022-00402-0","DOIUrl":"https://doi.org/10.1007/s40993-022-00402-0","url":null,"abstract":"Abstract This paper presents a fast method to compute algebraic norms of integral elements of smooth-degree cyclotomic fields, and, more generally, smooth-degree Galois number fields with commutative Galois groups. The typical scenario arising in S -unit searches (for, e.g., class-group computation) is computing a $$Theta (nlog n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Θ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -bit norm of an element of weight $$n^{1/2+o(1)}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> in a degree- n field; this method then uses $$n(log n)^{3+o(1)}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> bit operations. An $$n(log n)^{O(1)}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> operation count was already known in two easier special cases: norms from power-of-2 cyclotomic fields via towers of power-of-2 cyclotomic subfields, and norms from multiquadratic fields via towers of multiquadratic subfields. This paper handles more general Abelian fields by identifying tower-compatible integral bases supporting fast multiplication; in particular, there is a synergy between tower-compatible Gauss-period integral bases and a fast-multiplication idea from Rader. As a baseline, this paper also analyzes various standard norm-computation techniques that apply to arbitrary number fields, concluding that all of these techniques use at least $$n^2(log n)^{2+o(1)}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> bit operations in the same scenario, even with fast subroutines for continued fractions and for complex FFTs. Compared to this baseline, algorithms dedicated to smooth-degree Abelian fields find each norm $","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"116 23","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136491","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}

引用次数: 1

Reciprocity and the kernel of Dedekind sums 互易性与Dedekind和的核

Research in Number Theory Pub Date : 2023-11-08 DOI: 10.1007/s40993-023-00484-4

Alexis LaBelle, Emily Van Bergeyk, Matthew P. Young

引用次数: 2