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Counting integer polynomials with several roots of maximal modulus 计算有多个最大模根的整数多项式
arXiv - MATH - Number Theory Pub Date : 2024-09-13 DOI: arxiv-2409.08625
Artūras Dubickas, Min Sha
{"title":"Counting integer polynomials with several roots of maximal modulus","authors":"Artūras Dubickas, Min Sha","doi":"arxiv-2409.08625","DOIUrl":"https://doi.org/arxiv-2409.08625","url":null,"abstract":"In this paper, for positive integers $H$ and $k leq n$, we obtain some\u0000estimates on the cardinality of the set of monic integer polynomials of degree\u0000$n$ and height bounded by $H$ with exactly $k$ roots of maximal modulus. These\u0000include lower and upper bounds in terms of $H$ for fixed $k$ and $n$. We also\u0000count reducible and irreducible polynomials in that set separately. Our results\u0000imply, for instance, that the number of monic integer irreducible polynomials\u0000of degree $n$ and height at most $H$ whose all $n$ roots have equal moduli is\u0000approximately $2H$ for odd $n$, while for even $n$ there are more than\u0000$H^{n/8}$ of such polynomials.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259922","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: 0
A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus 希钦小面上堆叠的 p$-adic 黎曼-希尔伯特对应关系
arXiv - MATH - Number Theory Pub Date : 2024-09-13 DOI: arxiv-2409.08785
Yudong Liu, Chenglong Ma, Xiecheng Nie, Xiaoyu Qu, Yupeng Wang
{"title":"A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus","authors":"Yudong Liu, Chenglong Ma, Xiecheng Nie, Xiaoyu Qu, Yupeng Wang","doi":"arxiv-2409.08785","DOIUrl":"https://doi.org/arxiv-2409.08785","url":null,"abstract":"Let $C$ be an algebraically closed perfectoid field over $Qp$ with the ring\u0000of integer $calO_C$ and the infinitesimal thickening $Ainf$. Let $frakX$ be\u0000a smooth formal scheme over $calO_C$ with a fixed smooth lifting $wtx$ over\u0000$Ainf$. Let $X$ be the generic fiber of $frakX$ and $wtX$ be its lifting\u0000over $BdRp$ induced by $wtx$. Let $MIC_r(wtX)^{{rm H-small}}$ and\u0000$rLrS_r(X,BBdRp)^{{rm H-small}}$ be the $v$-stacks of rank-$r$\u0000Hitchin-small integrable connections on $wtX_{et}$ and $BBdRp$-local systems\u0000on $X_{v}$, respectively. In this paper, we establish an equivalence between\u0000this two stacks by introducing a new period sheaf with connection\u0000$(calObB_{dR,pd}^+,rd)$ on $X_{v}$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"214 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259977","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: 0
On the number of irreducible factors with a given multiplicity in function fields 论函数场中具有给定乘数的不可还原因子数
arXiv - MATH - Number Theory Pub Date : 2024-09-13 DOI: arxiv-2409.08559
Sourabhashis Das, Ertan Elma, Wentang Kuo, Yu-Ru Liu
{"title":"On the number of irreducible factors with a given multiplicity in function fields","authors":"Sourabhashis Das, Ertan Elma, Wentang Kuo, Yu-Ru Liu","doi":"arxiv-2409.08559","DOIUrl":"https://doi.org/arxiv-2409.08559","url":null,"abstract":"Let $k geq 1$ be a natural number and $f in mathbb{F}_q[t]$ be a monic\u0000polynomial. Let $omega_k(f)$ denote the number of distinct monic irreducible\u0000factors of $f$ with multiplicity $k$. We obtain asymptotic estimates for the\u0000first and the second moments of $omega_k(f)$ with $k geq 1$. Moreover, we\u0000prove that the function $omega_1(f)$ has normal order $log (text{deg}(f))$\u0000and also satisfies the ErdH{o}s-Kac Theorem. Finally, we prove that the\u0000functions $omega_k(f)$ with $k geq 2$ do not have normal order.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"84 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259923","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: 0
An Attack on $p$-adic Lattice Public-key Cryptosystems and Signature Schemes 对 p$-adic 网格公钥密码系统和签名方案的攻击
arXiv - MATH - Number Theory Pub Date : 2024-09-13 DOI: arxiv-2409.08774
Chi Zhang
{"title":"An Attack on $p$-adic Lattice Public-key Cryptosystems and Signature Schemes","authors":"Chi Zhang","doi":"arxiv-2409.08774","DOIUrl":"https://doi.org/arxiv-2409.08774","url":null,"abstract":"Lattices have many significant applications in cryptography. In 2021, the\u0000$p$-adic signature scheme and public-key encryption cryptosystem were\u0000introduced. They are based on the Longest Vector Problem (LVP) and the Closest\u0000Vector Problem (CVP) in $p$-adic lattices. These problems are considered to be\u0000challenging and there are no known deterministic polynomial time algorithms to\u0000solve them. In this paper, we improve the LVP algorithm in local fields. The\u0000modified LVP algorithm is a deterministic polynomial time algorithm when the\u0000field is totally ramified and $p$ is a polynomial in the rank of the input\u0000lattice. We utilize this algorithm to attack the above schemes so that we are\u0000able to forge a valid signature of any message and decrypt any ciphertext.\u0000Although these schemes are broken, this work does not mean that $p$-adic\u0000lattices are not suitable in constructing cryptographic primitives. We propose\u0000some possible modifications to avoid our attack at the end of this paper.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269318","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: 0
Creating a dynamic database of finite groups 创建有限组动态数据库
arXiv - MATH - Number Theory Pub Date : 2024-09-13 DOI: arxiv-2409.09189
Lewis Combes, John W. Jones, Jennifer Paulhus, David Roe, Manami Roy, Sam Schiavone
{"title":"Creating a dynamic database of finite groups","authors":"Lewis Combes, John W. Jones, Jennifer Paulhus, David Roe, Manami Roy, Sam Schiavone","doi":"arxiv-2409.09189","DOIUrl":"https://doi.org/arxiv-2409.09189","url":null,"abstract":"A database of abstract groups has been added to the L-functions and Modular\u0000Forms Database (LMFDB), available at https://www.lmfdb.org/Groups/Abstract/. We\u0000discuss the functionality of the database and what makes it distinct from other\u0000available databases of abstract groups. We describe solutions to mathematical\u0000problems we encountered while creating the database, as well as connections\u0000between the abstract groups database and other collections of objects in the\u0000LMFDB.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259917","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: 0
Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms 有限域上的无性变种与交换内态代数：理论与算法
arXiv - MATH - Number Theory Pub Date : 2024-09-13 DOI: arxiv-2409.08865
Jonas Bergström, Valentijn Karemaker, Stefano Marseglia
{"title":"Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms","authors":"Jonas Bergström, Valentijn Karemaker, Stefano Marseglia","doi":"arxiv-2409.08865","DOIUrl":"https://doi.org/arxiv-2409.08865","url":null,"abstract":"We give a categorical description of all abelian varieties with commutative\u0000endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny\u0000class in terms of pairs consisting of a fractional $mathbb Z[pi,q/pi]$-ideal\u0000and a fractional $Wotimes_{mathbb Z_p} mathbb Z_p[pi,q/pi]$-ideal, with\u0000$pi$ the Frobenius endomorphism and $W$ the ring of integers in an unramified\u0000extension of $mathbb Q_p$ of degree $a$. The latter ideal should be compatible\u0000at $p$ with the former and stable under the action of a semilinear Frobenius\u0000(and Verschiebung) operator; it will be the Dieudonn'e module of the\u0000corresponding abelian variety. Using this categorical description we create\u0000effective algorithms to compute isomorphism classes of these objects and we\u0000produce many new examples exhibiting exotic patterns.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"214 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259919","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: 0
Eventual tightness of projective dimension growth bounds: quadratic in the degree 投影维度增长边界的最终严密性：度数的二次方
arXiv - MATH - Number Theory Pub Date : 2024-09-13 DOI: arxiv-2409.08776
Raf Cluckers, Itay Glazer
{"title":"Eventual tightness of projective dimension growth bounds: quadratic in the degree","authors":"Raf Cluckers, Itay Glazer","doi":"arxiv-2409.08776","DOIUrl":"https://doi.org/arxiv-2409.08776","url":null,"abstract":"In projective dimension growth results, one bounds the number of rational\u0000points of height at most $H$ on an irreducible hypersurface in $mathbb P^n$ of\u0000degree $d>3$ by $C(n)d^2 H^{n-1}(log H)^{M(n)}$, where the quadratic\u0000dependence in $d$ has been recently obtained by Binyamini, Cluckers and Kato in\u00002024 [1]. For these bounds, it was already shown by Castryck, Cluckers,\u0000Dittmann and Nguyen in 2020 [3] that one cannot do better than a linear\u0000dependence in $d$. In this paper we show that, for the mentioned projective\u0000dimension growth bounds, the quadratic dependence in $d$ is eventually tight\u0000when $n$ grows. More precisely the upper bounds cannot be better than\u0000$c(n)d^{2-2/n} H^{n-1}$ in general. Note that for affine dimension growth (for\u0000affine hypersurfaces of degree $d$, satisfying some extra conditions), the\u0000dependence on $d$ is also quadratic by [1], which is already known to be\u0000optimal by [3]. Our projective case thus complements the picture of tightness\u0000for dimension growth bounds for hypersurfaces.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259920","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: 0
On the Artin formalism for triple product $p$-adic $L$-functions: Chow--Heegner points vs. Heegner points 关于 $p$-adic $L$ 函数的三重乘 $p$-adic $L$ 函数的阿廷形式主义：周--黑格纳点与黑格纳点
arXiv - MATH - Number Theory Pub Date : 2024-09-13 DOI: arxiv-2409.08645
Kâzım Büyükboduk, Daniele Casazza, Aprameyo Pal, Carlos de Vera-Piquero
{"title":"On the Artin formalism for triple product $p$-adic $L$-functions: Chow--Heegner points vs. Heegner points","authors":"Kâzım Büyükboduk, Daniele Casazza, Aprameyo Pal, Carlos de Vera-Piquero","doi":"arxiv-2409.08645","DOIUrl":"https://doi.org/arxiv-2409.08645","url":null,"abstract":"Our main objective in this paper (which is expository for the most part) is\u0000to study the necessary steps to prove a factorization formula for a certain\u0000triple product $p$-adic $L$-function guided by the Artin formalism. The key\u0000ingredients are: a) the explicit reciprocity laws governing the relationship of\u0000diagonal cycles and generalized Heegner cycles to $p$-adic $L$-functions; b) a\u0000careful comparison of Chow--Heegner points and twisted Heegner points in Hida\u0000families, via formulae of Gross--Zagier type.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259921","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: 0
Uniform polynomial bounds on torsion from rational geometric isogeny classes 从有理几何同源类看扭转的均匀多项式边界
arXiv - MATH - Number Theory Pub Date : 2024-09-12 DOI: arxiv-2409.08214
Abbey Bourdon, Tyler Genao
{"title":"Uniform polynomial bounds on torsion from rational geometric isogeny classes","authors":"Abbey Bourdon, Tyler Genao","doi":"arxiv-2409.08214","DOIUrl":"https://doi.org/arxiv-2409.08214","url":null,"abstract":"In 1996, Merel showed there exists a function $Bcolon\u0000mathbb{Z}^+rightarrow mathbb{Z}^+$ such that for any elliptic curve $E/F$\u0000defined over a number field of degree $d$, one has the torsion group bound $#\u0000E(F)[textrm{tors}]leq B(d)$. Based on subsequent work, it is conjectured that\u0000one can choose $B$ to be polynomial in the degree $d$. In this paper, we show\u0000that such bounds exist for torsion from the family $mathcal{I}_{mathbb{Q}}$\u0000of elliptic curves which are geometrically isogenous to at least one rational\u0000elliptic curve. More precisely, we show that for each $epsilon>0$ there exists\u0000$c_epsilon>0$ such that for any elliptic curve $E/Fin\u0000mathcal{I}_{mathbb{Q}}$, one has [ # E(F)[textrm{tors}]leq\u0000c_epsiloncdot [F:mathbb{Q}]^{5+epsilon}. ] This generalizes prior work of\u0000Clark and Pollack, as well as work of the second author in the case of rational\u0000geometric isogeny classes.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203796","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: 0
Universal sums of generalized polygonal numbers of almost prime "length" 几乎是素数 "长度 "的广义多边形数的普遍和
arXiv - MATH - Number Theory Pub Date : 2024-09-12 DOI: arxiv-2409.07895
Soumyarup Banerjee, Ben Kane, Daejun Kim
{"title":"Universal sums of generalized polygonal numbers of almost prime \"length\"","authors":"Soumyarup Banerjee, Ben Kane, Daejun Kim","doi":"arxiv-2409.07895","DOIUrl":"https://doi.org/arxiv-2409.07895","url":null,"abstract":"In this paper, we consider sums of three generalized $m$-gonal numbers whose\u0000parameters are restricted to integers with a bounded number of prime divisors.\u0000With some restrictions on $m$ modulo $30$, we show that a density one set of\u0000integers is represented as such a sum, where the parameters are restricted to\u0000have at most 6361 prime factors. Moreover, if the squarefree part of $f_m(n)$\u0000is sufficiently large, then $n$ is represented as such a sum, where $f_m(n)$ is\u0000a natural linear function in $n$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203799","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: 0
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