scholarly journals ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

2020 ◽  
pp. 1-10
Author(s):  
CLEMENS FUCHS ◽  
SEBASTIAN HEINTZE
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Linear Recurrences ◽  
Field Case ◽  
Power Sum ◽  
Linear Recurrence Sequence

Abstract Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

scholarly journals On the Selmer groups of abelian varieties over function fields of characteristic p > 0

2009 ◽  
Vol 146 (1) ◽  
pp. 23-43 ◽  
Author(s):  
TADASHI OCHIAI ◽  
FABIEN TRIHAN
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Abelian Varieties ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Characteristic P ◽  
Field Case ◽  
New Phenomena ◽  
Geometric Analogue

AbstractWe study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

FUNCTIONAL EQUATION OF CHARACTERISTIC ELEMENTS OF ABELIAN VARIETIES OVER FUNCTION FIELDS (ℓ ≠ p)

2014 ◽  
Vol 10 (03) ◽  
pp. 705-735
Author(s):  
APRAMEYO PAL
Keyword(s):  
Functional Equation ◽  
Number Field ◽  
Function Field ◽  
Functional Equations ◽  
Function Fields ◽  
Growth Behavior ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Selmer Group ◽  
Field Case

In this paper we apply methods from the number field case of Perrin-Riou [20] and Zábrádi [32] in the function field setup. In ℤℓ- and GL2-cases (ℓ ≠ p), we prove algebraic functional equations of the Pontryagin dual of Selmer group which give further evidence of the main conjectures of Iwasawa theory. We also prove some parity conjectures in commutative and non-commutative cases. As a consequence, we also get results on the growth behavior of Selmer groups in commutative and non-commutative extension of function fields.

scholarly journals Identities Involving the Fourth-Order Linear Recurrence Sequence

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1476 ◽  
Author(s):  
Lan Qi ◽  
Zhuoyu Chen
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Generating Function ◽  
Fourth Order ◽  
Power Series Expansion ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Order Linear ◽  
Elementary Methods ◽  
Linear Recurrence Sequence

In this paper, we introduce the fourth-order linear recurrence sequence and its generating function and obtain the exact coefficient expression of the power series expansion using elementary methods and symmetric properties of the summation processes. At the same time, we establish some relations involving Tetranacci numbers and give some interesting identities.

scholarly journals Some Convolution Formulae Related to the Second-Order Linear Recurrence Sequence

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 788 ◽  
Author(s):  
Zhuoyu Chen ◽  
Lan Qi
Keyword(s):  
Power Series Expansion ◽  
Second Order ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Order Linear ◽  
Symmetry Properties ◽  
Elementary Methods ◽  
Linear Recurrence Sequence ◽  
Convolution Formula ◽  
The Relationship

The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of the power series expansion of f x ( t ) for x ∈ R with elementary methods and symmetry properties. On the other hand, if we take some special values for a and b, not only can we obtain the convolution formula of some important polynomials, but also we can establish the relationship between polynomials and themselves. For example, we can find relationship between the Chebyshev polynomials and Legendre polynomials.

scholarly journals Chebyshev’s bias in function fields

2008 ◽  
Vol 144 (6) ◽  
pp. 1351-1374 ◽  
Author(s):  
Byungchul Cha
Keyword(s):  
Finite Field ◽  
Number Field ◽  
Central Limit ◽  
Polynomial Ring ◽  
Rational Number ◽  
Function Field ◽  
Sufficient Conditions ◽  
Monic Polynomial ◽  
Field Case ◽  
Necessary And Sufficient

AbstractWe study a function field analog of Chebyshev’s bias. Our results, as well as their proofs, are similar to those of Rubinstein and Sarnak in the case of the rational number field. Following Rubinstein and Sarnak, we introduce the grand simplicity hypothesis (GSH), a certain hypothesis on the inverse zeros of Dirichlet L-series of a polynomial ring over a finite field. Under this hypothesis, we investigate how primes, that is, irreducible monic polynomials in a polynomial ring over a finite field, are distributed in a given set of residue classes modulo a fixed monic polynomial. In particular, we prove under the GSH that, like the number field case, primes are biased toward quadratic nonresidues. Unlike the number field case, the GSH can be proved to hold in some cases and can be violated in some other cases. Also, under the GSH, we give the necessary and sufficient conditions for which primes are unbiased and describe certain central limit behaviors as the degree of modulus under consideration tends to infinity, all of which have been established in the number field case by Rubinstein and Sarnak.

scholarly journals Squarefree doubly primitive divisors in dynamical sequences

2017 ◽  
Vol 164 (3) ◽  
pp. 551-572 ◽  
Author(s):  
DRAGOS GHIOCA ◽  
KHOA D. NGUYEN ◽  
THOMAS J. TUCKER
Keyword(s):  
Number Field ◽  
Function Field ◽  
Field Case ◽  
Primitive Divisors ◽  
Finite Set ◽  
Function Field Case

AbstractLetKbe a number field or a function field of characteristic 0, letφ∈K(z) with deg(φ) ⩾ 2, and letα∈ ℙ1(K). LetSbe a finite set of places ofKcontaining all the archimedean ones and the primes whereφhas bad reduction. After excluding all the natural counterexamples, we define a subsetA(φ,α) of ℤ⩾0× ℤ>0and show that for all but finitely many (m,n) ∈A(φ,α) there is a prime 𝔭 ∉Ssuch that ord𝔭(φm+n(α)−φm(α)) = 1 andαhas portrait (m,n) under the action ofφmodulo 𝔭. This latter condition implies ord𝔭(φu+v(α)−φu(α)) ⩽ 0 for (u,v) ∈ ℤ⩾0× ℤ>0satisfyingu<morv<n. Our proof assumes a conjecture of Vojta for ℙ1× ℙ1in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of Ingram–Silverman, Faber–Granville and the authors.

scholarly journals The density of primes dividing a particular non-linear recurrence sequence

2016 ◽  
pp. 1-30
Author(s):  
Alexi Block Gorman ◽  
Tyler Genao ◽  
Heesu Hwang ◽  
Noam Kantor ◽  
Sarah Parsons ◽  
...  
Keyword(s):  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Non Linear ◽  
Linear Recurrence Sequence
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