We improve Mahler's lower bound for the Mahler measure in terms of the discriminant and degree for a specific class of polynomials: complex monic polynomials of degree $dgeq 2$ such that all roots with modulus greater than some fixed value $rgeq1$ occur in equal modulus pairs. We improve Mahler's exponent $frac{1}{2d-2}$ on the discriminant to $frac{1}{2d-3}$. Moreover, we show that this value is sharp, even when restricting to minimal polynomials of integral generators of a fixed not totally real number field. An immediate consequence of this new lower bound is an improved lower bound for integral generators of number fields, generalising a simple observation of Ruppert from imaginary quadratic to totally complex number fields of arbitrary degree.
{"title":"On Mahler’s inequality and small integral generators of totally complex number fields","authors":"Murray Child, Martin Widmer","doi":"10.4064/aa230601-18-9","DOIUrl":"https://doi.org/10.4064/aa230601-18-9","url":null,"abstract":"We improve Mahler's lower bound for the Mahler measure in terms of the discriminant and degree for a specific class of polynomials: complex monic polynomials of degree $dgeq 2$ such that all roots with modulus greater than some fixed value $rgeq1$ occur in equal modulus pairs. We improve Mahler's exponent $frac{1}{2d-2}$ on the discriminant to $frac{1}{2d-3}$. Moreover, we show that this value is sharp, even when restricting to minimal polynomials of integral generators of a fixed not totally real number field. An immediate consequence of this new lower bound is an improved lower bound for integral generators of number fields, generalising a simple observation of Ruppert from imaginary quadratic to totally complex number fields of arbitrary degree.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"70 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139351233","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}
B. Earp-Lynch, Bernadette Faye, E. Goedhart, I. Vukusic, Daniel P. Wisniewski
Let $t$ be any imaginary quadratic integer with $|t|geq 100$. We prove that the inequality [ |F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | leq 1 ] has only trivial solutions $(x,y)$ in integers of the same imaginary quadratic number field as $t$. Moreover, we prove results on the inequalities $|F_t(X,Y)| leq C|t|$ and $|F_t(X,Y)| leq |t|^{2 -varepsilon}$. These results follow from an approximation result that is based on the hypergeometric method. The proofs in this paper require a fair amount of computations, for which the code (in Sage) is provided.
设$t$为任意带$|t|geq 100$的虚二次整数。证明了不等式[ |F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | leq 1 ]在与$t$相同的虚二次数域的整数中只有平凡解$(x,y)$。此外，我们还证明了不等式$|F_t(X,Y)| leq C|t|$和$|F_t(X,Y)| leq |t|^{2 -varepsilon}$的结果。这些结果来自基于超几何方法的近似结果。本文中的证明需要相当数量的计算，为此提供了代码(在Sage中)。
{"title":"On a simple quartic family of Thue equations over imaginary quadratic number fields","authors":"B. Earp-Lynch, Bernadette Faye, E. Goedhart, I. Vukusic, Daniel P. Wisniewski","doi":"10.4064/aa230329-19-6","DOIUrl":"https://doi.org/10.4064/aa230329-19-6","url":null,"abstract":"Let $t$ be any imaginary quadratic integer with $|t|geq 100$. We prove that the inequality [ |F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | leq 1 ] has only trivial solutions $(x,y)$ in integers of the same imaginary quadratic number field as $t$. Moreover, we prove results on the inequalities $|F_t(X,Y)| leq C|t|$ and $|F_t(X,Y)| leq |t|^{2 -varepsilon}$. These results follow from an approximation result that is based on the hypergeometric method. The proofs in this paper require a fair amount of computations, for which the code (in Sage) is provided.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44168848","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}
We generalize results of Duke, Garcia, Hyde, Lutz and others on the distribution of sums of roots of unity related to Gaussian periods to obtain equidistribution of similar sums over zeros of arbitrary integral polynomials. We also interpret these results in terms of trace functions, and generalize them to higher rank trace functions.
{"title":"Ultra-short sums of trace functions","authors":"E. Kowalski, Th'eo Untrau","doi":"10.4064/aa230308-11-5","DOIUrl":"https://doi.org/10.4064/aa230308-11-5","url":null,"abstract":"We generalize results of Duke, Garcia, Hyde, Lutz and others on the distribution of sums of roots of unity related to Gaussian periods to obtain equidistribution of similar sums over zeros of arbitrary integral polynomials. We also interpret these results in terms of trace functions, and generalize them to higher rank trace functions.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41947090","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}
Fix two distinct odd primes $p$ and $q$. We study"$pne q$"Iwasawa theory in two different settings. Let $K$ be an imaginary quadratic field of class number 1 such that both $p$ and $q$ split in $K$. We show that under appropriate hypotheses, the $p$-part of the ideal class groups is bounded over finite subextensions of an anticyclotomic $mathbb{Z}_q$-extension of $K$. Let $F$ be a number field and let $A_{/F}$ be an abelian variety with $A[p]subseteq A(F)$. We give sufficient conditions for the $p$-part of the fine Selmer groups of $A$ over finite subextensions of a $mathbb{Z}_q$-extension of $F$ to stabilize.
{"title":"Growth of $p$-parts of ideal class groups and fine Selmer groups in $mathbb Z_q$-extensions with $pne q$","authors":"Debanjana Kundu, Antonio Lei","doi":"10.4064/aa220518-28-2","DOIUrl":"https://doi.org/10.4064/aa220518-28-2","url":null,"abstract":"Fix two distinct odd primes $p$ and $q$. We study\"$pne q$\"Iwasawa theory in two different settings. Let $K$ be an imaginary quadratic field of class number 1 such that both $p$ and $q$ split in $K$. We show that under appropriate hypotheses, the $p$-part of the ideal class groups is bounded over finite subextensions of an anticyclotomic $mathbb{Z}_q$-extension of $K$. Let $F$ be a number field and let $A_{/F}$ be an abelian variety with $A[p]subseteq A(F)$. We give sufficient conditions for the $p$-part of the fine Selmer groups of $A$ over finite subextensions of a $mathbb{Z}_q$-extension of $F$ to stabilize.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45647920","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}
We describe the higher even $K$-groups of the ring of integers of a number field in terms of the class groups of an appropriate extension of the number field in question. This is a natural extension of the previous work of Browkin, Keune and Kolster, who
{"title":"On the structure of even $K$-groups of rings of algebraic integers","authors":"Meng Fai Lim","doi":"10.4064/aa221029-25-7","DOIUrl":"https://doi.org/10.4064/aa221029-25-7","url":null,"abstract":"We describe the higher even $K$-groups of the ring of integers of a number field in terms of the class groups of an appropriate extension of the number field in question. This is a natural extension of the previous work of Browkin, Keune and Kolster, who","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135051529","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}
For positive integers $k$ and $n$ let $sigma _k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erdős and Kac asked whether, for every $k$, the number $alpha _k = sum _{ngeq 1} frac {sigma _k(n)}{n!}$ is irrational. It is known uncond
{"title":"The irrationality of a divisor function series of Erdős and Kac","authors":"Kyle Pratt","doi":"10.4064/aa220927-1-9","DOIUrl":"https://doi.org/10.4064/aa220927-1-9","url":null,"abstract":"For positive integers $k$ and $n$ let $sigma _k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erdős and Kac asked whether, for every $k$, the number $alpha _k = sum _{ngeq 1} frac {sigma _k(n)}{n!}$ is irrational. It is known uncond","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135559808","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}
Let $F$ be a number field, and let $pi_1$ and $pi_2$ be distinct unitary cuspidal automorphic representations of $operatorname{GL}_{n_1}(mathbb{A}_F)$ and $operatorname{GL}_{n_2}(mathbb{A}_F)$ respectively. In this paper, we derive new lower bounds for the Rankin-Selberg $L$-function $L(s, pi_1 times widetilde{pi}_2)$ along the edge $Re s = 1$ of the critical strip in the $t$-aspect. The corresponding zero-free region for $L(s, pi_1 times widetilde{pi}_2)$ is also determined.
{"title":"Lower bounds for Rankin–Selberg $L$-functions on the edge of the critical strip","authors":"Qiao Zhang","doi":"10.4064/aa221111-14-7","DOIUrl":"https://doi.org/10.4064/aa221111-14-7","url":null,"abstract":"Let $F$ be a number field, and let $pi_1$ and $pi_2$ be distinct unitary cuspidal automorphic representations of $operatorname{GL}_{n_1}(mathbb{A}_F)$ and $operatorname{GL}_{n_2}(mathbb{A}_F)$ respectively. In this paper, we derive new lower bounds for the Rankin-Selberg $L$-function $L(s, pi_1 times widetilde{pi}_2)$ along the edge $Re s = 1$ of the critical strip in the $t$-aspect. The corresponding zero-free region for $L(s, pi_1 times widetilde{pi}_2)$ is also determined.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136260001","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":"Ramanujan–Sato series for $1/pi $","authors":"Tim Huber, Daniel Schultz, Dongxi Ye","doi":"10.4064/aa220621-19-12","DOIUrl":"https://doi.org/10.4064/aa220621-19-12","url":null,"abstract":"","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70440175","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}
. In this paper we give finiteness results for the shifted power values and polynomial values of Littlewood polynomials.
． 本文给出了Littlewood多项式的移幂值和多项式值的有限性结果。
{"title":"Diophantine equations for Littlewood polynomials","authors":"L. Hajdu, R. Tijdeman, N. Varga","doi":"10.4064/aa220912-3-11","DOIUrl":"https://doi.org/10.4064/aa220912-3-11","url":null,"abstract":". In this paper we give finiteness results for the shifted power values and polynomial values of Littlewood polynomials.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70440351","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":"Additive decomposition of signed primes","authors":"I. Ruzsa","doi":"10.4064/aa220429-17-11","DOIUrl":"https://doi.org/10.4064/aa220429-17-11","url":null,"abstract":"","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70440088","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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