{"title":"A uniform semi-local limit theorem along sets of multiples for sums of i.i.d. random variables","authors":"Michel J.G. Weber","doi":"10.7169/facm/2078","DOIUrl":"https://doi.org/10.7169/facm/2078","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140525759","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}
{"title":"On the class numbers of the $n$-th layers in the cyclotomic $mathbb{Z}_2$-extension of $mathbb{Q}(sqrt{5})$","authors":"Takuya Aoki","doi":"10.7169/facm/2058","DOIUrl":"https://doi.org/10.7169/facm/2058","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140520813","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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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 $n$.
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
N. Vani, D. Vamshee Krishna, Ch. Vijaya Kumar, B. Rath, K. Sanjay Kumar
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 $mathcal{P}$.
{"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":null,"pages":null},"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}
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