The Pintz–Steiger–Szemerédi estimate for intersective quadratic polynomials in function fields

2021 ◽  
pp. 1-50
Author(s):  
Guoquan Li
Keyword(s):  
Finite Field ◽  
Natural Number ◽  
Polynomial Ring ◽  
Function Fields ◽  
Difference Set ◽  
Quadratic Polynomial ◽  
Quadratic Polynomials ◽  
Number Formula ◽  
The Difference

Let [Formula: see text] be the polynomial ring over the finite field [Formula: see text] of [Formula: see text] elements. For a natural number [Formula: see text] let [Formula: see text] be the set of all polynomials in [Formula: see text] of degree less than [Formula: see text] Let [Formula: see text] be a quadratic polynomial over [Formula: see text] Suppose that [Formula: see text] is intersective, that is, which satisfies [Formula: see text] for any [Formula: see text] with [Formula: see text] where [Formula: see text] denotes the difference set of [Formula: see text] Let [Formula: see text] Suppose that [Formula: see text] and that the characteristic of [Formula: see text] is not divisible by 2. It is proved that [Formula: see text] for any [Formula: see text] where [Formula: see text] is a constant depending only on [Formula: see text] and [Formula: see text]

scholarly journals A GENERALIZATION OF ROTH'S THEOREM IN FUNCTION FIELDS

2009 ◽  
Vol 05 (07) ◽  
pp. 1149-1154 ◽  
Author(s):  
YU-RU LIU ◽  
CRAIG V. SPENCER
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Trivial Solution ◽  
Function Fields

Let 𝔽q[t] denote the polynomial ring over the finite field 𝔽q, and let [Formula: see text] denote the subset of 𝔽q[t] containing all polynomials of degree strictly less than N. For non-zero elements r1, …, rs of 𝔽q satisfying r1 + ⋯ + rs = 0, let [Formula: see text] denote the maximal cardinality of a set [Formula: see text] which contains no non-trivial solution of r1x1 + ⋯ + rsxs = 0 with xi ∈ A (1 ≤ i ≤ s). We prove that [Formula: see text].

Multidimensional Vinogradov-type Estimates in Function Fields

2014 ◽  
Vol 66 (4) ◽  
pp. 844-873 ◽  
Author(s):  
Wentang Kuo ◽  
Yu-Ru Liu ◽  
Xiaomei Zhao
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Function Fields ◽  
Circle Method ◽  
Asymptotic Formulas ◽  
Linear Spaces

AbstractLet 𝔽q[t] denote the polynomial ring over the finite field 𝔽q. We employ Wooley's new efficient congruencing method to prove certain multidimensional Vinogradov-type estimates in 𝔽q[t]. These results allow us to apply a variant of the circle method to obtain asymptotic formulas for a system connected to the problem about linear spaces lying on hypersurfaces defined over 𝔽q[t].

Regulators of an Infinite Family of the Simplest Quartic Function Fields

2017 ◽  
Vol 69 (3) ◽  
pp. 579-594 ◽  
Author(s):  
Jungyun Lee ◽  
Yoonjin Lee
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Polynomial Ring ◽  
Function Fields ◽  
Infinite Family ◽  
Class Numbers ◽  
Explicit Criterion ◽  
Divisor Class ◽  
The Family

AbstractWe explicitly find regulators of an infinite family {Lm} of the simplest quartic function fields with a parameter m in a polynomial ring [t], where is the finite field of order q with odd characteristic. In fact, this infinite family of the simplest quartic function fields are subfields of maximal real subfields of cyclotomic function fields having the same conductors. We obtain a lower bound on the class numbers of the family {Lm} and some result on the divisibility of the divisor class numbers of cyclotomic function fields that contain {Lm} as their subfields. Furthermore, we find an explicit criterion for the characterization of splitting types of all the primes of the rational function field (t) in {Lm}.

A method of version merging for computer-aided design files based on feature extraction

2010 ◽  
Vol 225 (2) ◽  
pp. 463-471 ◽  
Author(s):  
C Sun ◽  
D Guo ◽  
H Gao ◽  
L Zou ◽  
H Wang
Keyword(s):  
Feature Extraction ◽  
Computer Aided Design ◽  
Application Programming Interface ◽  
Difference Set ◽  
Computer Aided ◽  
Application Programming ◽  
Feature Difference ◽  
The Difference ◽  
Aided Design ◽  
The Given

In order to manage the version files and maintain the latest version of the computer-aided design (CAD) files in asynchronous collaborative systems, one method of version merging for CAD files is proposed to resolve the problem based on feature extraction. First of all, the feature information is extracted based on the feature attribute of CAD files and stored in a XML feature file. Then, analyse the feature file, and the feature difference set is obtained by the given algorithm. Finally, the merging result of the difference set and the master files with application programming interface (API) interface functions is achieved, and then the version merging of CAD files is also realized. The application in Catia validated that the proposed method is feasible and valuable in engineering.

Generalized Artin'S Conjecture for Primitive Roots and Cyclicity Mod of Elliptic Curves Over Function Fields

1995 ◽  
Vol 38 (2) ◽  
pp. 167-173 ◽  
Author(s):  
David A. Clark ◽  
Masato Kuwata
Keyword(s):  
Finite Field ◽  
Prime Ideal ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Function Fields ◽  
Artin's Conjecture ◽  
Primitive Roots

AbstractLet k = Fq be a finite field of characteristic p with q elements and let K be a function field of one variable over k. Consider an elliptic curve E defined over K. We determine how often the reduction of this elliptic curve to a prime ideal is cyclic. This is done by generalizing a result of Bilharz to a more general form of Artin's primitive roots problem formulated by R. Murty.

THE EXCEPTIONAL SET IN THE POLYNOMIAL GOLDBACH PROBLEM

2011 ◽  
Vol 07 (03) ◽  
pp. 579-591 ◽  
Author(s):  
PAUL POLLACK
Keyword(s):  
Asymptotic Formula ◽  
Finite Field ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Natural Numbers ◽  
Exceptional Set ◽  
Goldbach Problem

For each natural number N, let R(N) denote the number of representations of N as a sum of two primes. Hardy and Littlewood proposed a plausible asymptotic formula for R(2N) and showed, under the assumption of the Riemann Hypothesis for Dirichlet L-functions, that the formula holds "on average" in a certain sense. From this they deduced (under ERH) that all but Oϵ(x1/2+ϵ) of the even natural numbers in [1, x] can be written as a sum of two primes. We generalize their results to the setting of polynomials over a finite field. Owing to Weil's Riemann Hypothesis, our results are unconditional.

On a conjecture about the quasirecognition by prime graph of some finite simple groups

2019 ◽  
Vol 18 (04) ◽  
pp. 1950070
Author(s):  
Ali Mahmoudifar
Keyword(s):  
Computer Program ◽  
Natural Number ◽  
Simple Group ◽  
Prime Number ◽  
Prime Power ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Number Formula ◽  
Finite Linear

It is proved that some finite simple groups are quasirecognizable by prime graph. In [A. Mahmoudifar and B. Khosravi, On quasirecognition by prime graph of the simple groups [Formula: see text] and [Formula: see text], J. Algebra Appl. 14(1) (2015) 12pp], the authors proved that if [Formula: see text] is a prime number and [Formula: see text], then there exists a natural number [Formula: see text] such that for all [Formula: see text], the simple group [Formula: see text] (where [Formula: see text] is a linear or unitary simple group) is quasirecognizable by prime graph. Also[Formula: see text] in that paper[Formula: see text] the author posed the following conjecture: Conjecture. For every prime power [Formula: see text] there exists a natural number [Formula: see text] such that for all [Formula: see text] the simple group [Formula: see text] is quasirecognizable by prime graph. In this paper [Formula: see text] as the main theorem we prove that if [Formula: see text] is a prime power and satisfies some especial conditions [Formula: see text] then there exists a number [Formula: see text] associated to [Formula: see text] such that for all [Formula: see text] the finite linear simple group [Formula: see text] is quasirecognizable by prime graph. Finally [Formula: see text] by a calculation via a computer program [Formula: see text] we conclude that the above conjecture is valid for the simple group [Formula: see text] where [Formula: see text] [Formula: see text] is an odd number and [Formula: see text].

scholarly journals Number fields and function fields: coalescences, contrasts and emerging applications

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140315 ◽  
Author(s):  
J. P. Keating ◽  
Z. Rudnick ◽  
T. D. Wooley
Keyword(s):  
Finite Field ◽  
Function Fields ◽  
Analytic Number Theory ◽  
Number Fields ◽  
Past Half Century ◽  
Recent Developments ◽  
Substantial Progress ◽  
And Function ◽  
Past Half ◽  
Field Setting

The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. We here introduce the papers published in this Theo Murphy meeting issue, where some of the recent developments are explained.

K3 of truncated polynomial rings over fields of characteristic two

1987 ◽  
Vol 101 (3) ◽  
pp. 509-521 ◽  
Author(s):  
Janet Aisbett ◽  
Victor Snaith
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Polynomial Rings ◽  
Witt Vectors ◽  
Characteristic Two

Write F for the finite field, , having 2m elements. Let W2(F) denote the Witt vectors of length two over F (for a definition, see [4] or [10], §10). Write F(q) for the truncated polynomial ring, F[t]/(tq).

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