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On the monogenicity of power-compositional Shanks polynomials 幂-复合Shanks多项式的单性性
FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI Pub Date : 2023-09-15 DOI: 10.7169/facm/2104
Lenny Jones
{"title":"On the monogenicity of power-compositional Shanks polynomials","authors":"Lenny Jones","doi":"10.7169/facm/2104","DOIUrl":"https://doi.org/10.7169/facm/2104","url":null,"abstract":"Let $f(x)in mathbb{Z}[x]$ be a monic polynomial of degree $N$ that is irreducible over $mathbb{Q}$. We say $f(x)$ is emph{monogenic} if $Theta={1,theta,theta^2,ldots ,theta^{N-1}}$ is a basis for the ring of integers $mathbb{Z}_K$ of $K=mathbb{Q}(theta)$, where $f(theta)=0$. If $Theta$ is not a basis for $mathbb{Z}_K$, we say that $f(x)$ is emph{non-monogenic}.Let $kge 1$ be an integer, and let $(U_n)$be the sequence defined by [U_0=U_1=0,qquad U_2=1 qquad text{and}qquad U_n=kU_{n-1}+(k+3)U_{n-2}+U_{n-3} qquad text{for $nge 3$}.] It is well known that $(U_n)$ is periodic modulo any integer $mge 2$, and we let $pi(m)$ denote the length of this period. We define a emph{$k$-Shanks prime} to be a prime $p$ such that $pi(p^2)=pi(p)$. Let $mathcal{S}_k(x)=x^{3}-kx^{2}-(k+3)x-1$ and $mathcal{D}=(k^2+3k+9)/gcd(3,k)^2$. Suppose that $knot equiv 3 pmod{9}$ and that $mathcal{D}$ is squarefree. In this article, we prove that $p$ is a $k$-Shanks prime if and only if $mathcal{S}_k(x^p)$ is non-monogenic, for any prime $p$ such that $mathcal{S}_k(x)$ is irreducible in $mathbb{F}_p[x]$. Furthermore, we show that $mathcal{S}_k(x^p)$ is monogenic for any prime divisor $p$ of $mathcal{D}$. These results extend previous work of the author on $k$-Wall-Sun-Sun primes.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135394108","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
Moments of Gaussian hypergeometric functions over finite fields 有限域上高斯超几何函数的矩
FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI Pub Date : 2023-09-15 DOI: 10.7169/facm/2088
Ankan Pal, Bidisha Roy, Mohammad Sadek
{"title":"Moments of Gaussian hypergeometric functions over finite fields","authors":"Ankan Pal, Bidisha Roy, Mohammad Sadek","doi":"10.7169/facm/2088","DOIUrl":"https://doi.org/10.7169/facm/2088","url":null,"abstract":"We prove explicit formulas for certain first and second moment sums of families of Gaussian hypergeometric functions $_{n+1}F_n$, $nge 1$, over finite fields with $q$ elements where $q$ is an odd prime. This enables us to find an estimate for the value $_6F_5(1)$. In addition, we evaluate certain second moments of traces of the family of Clausen elliptic curves in terms of the value $_3F_2(-1)$. These formulas also allow us to express the product of certain $_2F_1$ and $_{n+1}F_n$ functions in terms of finite field Appell series which generalizes current formulas for products of $_2F_1$ functions. We finally give closed form expressions for sums of Gaussian hypergeometric functions defined using different multiplicative characters.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135353867","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 arithmetic of polynomials with coefficients in Mordell-Weil type groups modell - weil型群中带系数多项式的算法
FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI Pub Date : 2023-09-15 DOI: 10.7169/facm/2105
Stefan Barańczuk
{"title":"On the arithmetic of polynomials with coefficients in Mordell-Weil type groups","authors":"Stefan Barańczuk","doi":"10.7169/facm/2105","DOIUrl":"https://doi.org/10.7169/facm/2105","url":null,"abstract":"In this paper we prove the Hasse principle for polynomials with coefficients in Mordell-Weil type groups over number fields. Examples of such groups are (1) the groups of $S$-units in a number field, (2) abelian varieties with trivial ring of endomorphisms, and (3) odd algebraic $K$-theory groups.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":"201 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135352768","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
Polynomials realizing images of Galois representations of an elliptic curve 实现椭圆曲线图像伽罗瓦表示的多项式
FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI Pub Date : 2023-09-15 DOI: 10.7169/facm/2106
Zoé Yvon
{"title":"Polynomials realizing images of Galois representations of an elliptic curve","authors":"Zoé Yvon","doi":"10.7169/facm/2106","DOIUrl":"https://doi.org/10.7169/facm/2106","url":null,"abstract":"The aim of the inverse Galois problem is to find extensions of a given field whose Galois group is isomorphic to a given group. In this article, we are interested in subgroups of $mathrm{GL}_2(mathbb{Z}/nmathbb{Z})$ where $n$ is an integer. We know that, in general, we can realize these groups as the Galois group of a given number field, using the torsion points on an elliptic curve. Specifically, a theorem of Reverter and Vila gives, for each prime $n$, a polynomial, depending on an elliptic curve, whose Galois group is $mathrm{GL}_2(mathbb{Z}/nmathbb{Z})$. In this article, we generalize this theorem in several directions, in particular for $n$ not necessarily prime. We also determine a minimum for the valuations of the coefficients of the polynomials arising in our construction, depending only on $n$.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":"355 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135394106","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
Bounding the number of lattice pointsnear a convex curve by curvature 用曲率限定凸曲线附近的点阵数目
FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI Pub Date : 2023-01-01 DOI: 10.7169/facm/2087
Ralph Howard, Ognian Trifonov
{"title":"Bounding the number of lattice pointsnear a convex curve by curvature","authors":"Ralph Howard, Ognian Trifonov","doi":"10.7169/facm/2087","DOIUrl":"https://doi.org/10.7169/facm/2087","url":null,"abstract":"We prove explicitbounds on the number of lattice points on or near a convex curve in termsof geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our estimates hold for lattices more general than the usual lattice ofintegral points in the plane.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":"73 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":"135106320","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
Bounds for the smallest integral solutionof Pell equation over a number field Pell方程在数域上的最小积分解的界
FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI Pub Date : 2023-01-01 DOI: 10.7169/facm/2095
Paraskevas Alvanos, Dimitrios Poulakis
{"title":"Bounds for the smallest integral solutionof Pell equation over a number field","authors":"Paraskevas Alvanos, Dimitrios Poulakis","doi":"10.7169/facm/2095","DOIUrl":"https://doi.org/10.7169/facm/2095","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":"1 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":"135107400","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
The variation of general Fourier coefficients 一般傅立叶系数的变化
IF 0.5
FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI Pub Date : 2023-01-01 DOI: 10.7169/facm/2002
V. Tsagareishvili
{"title":"The variation of general Fourier coefficients","authors":"V. Tsagareishvili","doi":"10.7169/facm/2002","DOIUrl":"https://doi.org/10.7169/facm/2002","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44332845","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
Coefficient bound associated with certain Hankel determinants and Zalcman conjecturefor a subfamily of multivalent bounded turning functions 多价有界翻转函数亚族与某些Hankel行列式相关的系数界和Zalcman猜想
FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI Pub Date : 2023-01-01 DOI: 10.7169/facm/2076
N. Vani, D. Vamshee Krishna, Ch. Vijaya Kumar, B. Rath, K. Sanjay Kumar
{"title":"Coefficient bound associated with certain Hankel determinants and Zalcman conjecturefor a subfamily of multivalent bounded turning functions","authors":"N. Vani, D. Vamshee Krishna, Ch. Vijaya Kumar, B. Rath, K. Sanjay Kumar","doi":"10.7169/facm/2076","DOIUrl":"https://doi.org/10.7169/facm/2076","url":null,"abstract":"In this paper, we introduce certain subfamily of $p$-valent analytic functions of bounded turning for which we estimate best possible upper bound to certain generalised second Hankel determinant, the Zalcman conjecture and an upper bound to the third, fourth Hankel determinants. Further, we investigate an upper bound for third and fourth Hankel determinants with respect to two-fold and three-fold symmetric functions for the same class. The practical tools applied in the derivation of our main results are the coefficient inequalities of the Carathéodory class $mathcal{P}$.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":"9 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":"135107183","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
Additive properties of the evil and odious numbers and similar sequences 恶数、恶数及相似数列的加性性质
FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI Pub Date : 2023-01-01 DOI: 10.7169/facm/2108
Jean-Paul Allouche, Jeffrey Shallit
{"title":"Additive properties of the evil and odious numbers and similar sequences","authors":"Jean-Paul Allouche, Jeffrey Shallit","doi":"10.7169/facm/2108","DOIUrl":"https://doi.org/10.7169/facm/2108","url":null,"abstract":"First we reprove two results in additive number theorydue to Dombi and Chen & Wang, respectively, on the number ofrepresentations of $n$ as the sum of two odious or evil numbers, using techniques from automata theory and logic. We also use this technique to prove a new result aboutthe numbers represented by five summands. Furthermore, we prove some new results on the tenfold sums of the evil and odious numbers, as well as $k$-fold sums of similar sequences of integers, by using techniques of analytic number theory involving trigonometric sums associated with the $pm 1$ characteristic sequences of these integers.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":"4 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":"135107188","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
The number of solutionsto the trinomial Thue equation 三叉方程解的个数
FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI Pub Date : 2023-01-01 DOI: 10.7169/facm/2093
Greg Knapp
{"title":"The number of solutionsto the trinomial Thue equation","authors":"Greg Knapp","doi":"10.7169/facm/2093","DOIUrl":"https://doi.org/10.7169/facm/2093","url":null,"abstract":"In this paper, we study the number of integer pair solutions to the equation $|F(x,y)| = 1$ where $F(x,y) in Z[x,y]$ is an irreducible (over $Z$) binary form with degree $n geq 3$ and exactly three nonzero summands. In particular, we improve Thomas' explicit upper bounds on the number of solutions to this equation (see [13]). For instance, when $n geq 219$, we show that there are no more than 32 integer pair solutions to this equation when $n$ is odd and no more than 40 integer pair solutions to this equation when $n$ is even, an improvement on Thomas' work in [13], where he shows that there are no more than 38 such solutions when $n$ is odd and no more than 48 such solutions when $n$ is even.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":"4 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":"135106313","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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