scholarly journals On the set of distances between two sets over finite fields

2006 ◽  
Vol 2006 ◽  
pp. 1-5 ◽  
Author(s):  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Exponential Sums

We use bounds of exponential sums to derive new lower bounds on the number of distinct distances between all pairs of points(x,y)∈×ℬfor two given sets,ℬ∈Fqn, whereFqis a finite field ofqelements andn≥1is an integer.

scholarly journals On some generalisations of the Erdős distance problem over finite fields

2006 ◽  
Vol 73 (2) ◽  
pp. 285-292 ◽  
Author(s):  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Exponential Sums ◽  
Distance Problem ◽  
Erdős Distance Problem ◽  
Erdos Distance Problem

We use exponential sums to obtain new lower bounds on the number of distinct distances defined by all pairs of points (a, b) ∈ A × B for two given sets where is a finite field of q elements and n ≥ 1 is an integer.

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

scholarly journals Characterization and Enumeration of Good Punctured Polynomials over Finite Fields

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Somphong Jitman ◽  
Aunyarut Bunyawat ◽  
Supanut Meesawat ◽  
Arithat Thanakulitthirat ◽  
Napat Thumwanit
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Complete Characterization ◽  
Binary Field ◽  
Polynomials Over Finite Fields

A family of good punctured polynomials is introduced. The complete characterization and enumeration of such polynomials are given over the binary fieldF2. Over a nonbinary finite fieldFq, the set of good punctured polynomials of degree less than or equal to2are completely determined. Forn≥3, constructive lower bounds of the number of good punctured polynomials of degreenoverFqare given.

The Distribution of Zeros of Quadratic Forms over Finite Fields

2015 ◽  
Vol 7 (2) ◽  
pp. 18
Author(s):  
Ali H. Hakami
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Quadratic Forms ◽  
Distribution Of Zeros ◽  
Linearly Independent ◽  
Set Of Points

Let $m$ be a positive integer with $m < p/2$ and $p$ is a prime. Let $\mathbb{F}_q$ be the finite field in $q = p^f$ elements, $Q({\mathbf{x}})$ be a nonsinqular quadratic form over $\mathbb{F}_q$ with $q$ odd, $V$ be the set of points in $\mathbb{F}_q^n$ satisfying the equation $Q({\mathbf{x}}) = 0$ in which the variables are restricted to a box of points of the type\[\mathcal{B}(m) = \left\{ {{\mathbf{x}} \in \mathbb{F}_q^n \left| {x_i  = \sum\limits_{j = 1}^f {x_{ij} \xi _j } ,\;\left| {x_{ij} } \right| < m,\;1 \leqslant i \leqslant n,\;1 \leqslant j \leqslant f} \right.} \right\},\]where $\xi _1 , \ldots ,\xi _f$ is a basis for $\mathbb{F}_q$ over $\mathbb{F}_p$ and $n > 2$ even. Set $\Delta  = \det Q$ such that $\chi \left( {( - 1)^{n/2} \Delta } \right) = 1.$ We shall motivate work of (Cochrane, 1986) to obtain lower bounds on $m,$ size of the box $\mathcal{B},$ so that $\mathcal{B} \cap V$ is nonempty. For this we show that the box $\mathcal{B}(m)$ contains a zero of $Q({\mathbf{x}})$ provided that $m \geqslant p^{1/2}.$ We also show that the box $\mathcal{B}(m)$ contains $n$ linearly independent zeros of $Q({\mathbf{x}})$ provided that $m \geqslant 2^{n/2} p^{1/2} .$

DEGREE MATRICES AND ESTIMATES FOR EXPONENTIAL SUMS OF POLYNOMIALS OVER FINITE FIELDS

2013 ◽  
Vol 12 (07) ◽  
pp. 1350030 ◽  
Author(s):  
WEI CAO
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Exponential Sums ◽  
Elementary Approach ◽  
Multivariate Polynomial ◽  
Elementary Divisors ◽  
Polynomials Over Finite Fields ◽  
Degree Matrix

Let f be a multivariate polynomial over a finite field and its degree matrix be composed of the degree vectors appearing in f. In this paper, we provide an elementary approach to estimating the exponential sums of the polynomials with positive square degree matrices in terms of the elementary divisors of the degree matrices.

Convolution and Equidistribution

2012 ◽  
Author(s):  
Nicholas M. Katz
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Mellin Transform ◽  
Exponential Sums ◽  
Multiplicative Group ◽  
Input Function ◽  
Point Of View ◽  
Basic Question ◽  
Mellin Transforms ◽  
The Subject

This book explores an important aspect of number theory—the theory of exponential sums over finite fields and their Mellin transforms—from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject. The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book's main theorem, using a mixture of geometric, categorical, and group-theoretic methods. By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject.

scholarly journals The Cardinality of the Set of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

2017 ◽  
Vol 9 (2) ◽  
pp. 56
Author(s):  
Yasanthi Kottegoda ◽  
Robert Fitzgerald
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Upper Bound ◽  
Cyclic Codes ◽  
Quadratic Residue ◽  
Upper And Lower Bounds ◽  
Linear Recurring Sequences ◽  
Irreducible Cyclic Codes ◽  
Homogeneous Linear

Consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on the irreducible characteristic polynomial of degree $d$ and order $m$. We give upper and lower bounds, and in some cases the exact values of the cardinality of the set of zeros of the sequences within its least period. We also prove that the cyclotomy bound introduced here is the best upper bound as it is reached in infinitely many cases. In addition, the exact number of occurrences of zeros is determined using the correlation with irreducible cyclic codes when $(q^{d}-1)/ m$ follows the quadratic residue conditions and also when it has the form $q^{2a}-q^{a}+1$ where $a\in \mathbb{N}$.

A NOTE ON SOME CHARACTER SUMS OVER FINITE FIELDS

2015 ◽  
Vol 92 (1) ◽  
pp. 32-43
Author(s):  
XIWANG CAO ◽  
GUANGKUI XU
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Boolean Functions ◽  
Exponential Sums ◽  
Character Sums ◽  
Walsh Spectrum

In this paper, we present a decomposition of the elements of a finite field and illustrate the efficiency of this decomposition in evaluating some specific exponential sums over finite fields. The results can be employed in determining the Walsh spectrum of some Boolean functions.

Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields

2012 ◽  
Vol 55 (2) ◽  
pp. 418-423 ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Bilinear Form ◽  
Symmetric Bilinear Form

AbstractGiven a positive integern, a finite fieldofqelements (qodd), and a non-degenerate symmetric bilinear formBon, we determine the largest possible cardinality of pairwiseB-orthogonal subsets, that is, for any two vectorsx,y∈ Ε, one hasB(x,y) = 0.

scholarly journals Short Kloosterman Sums for Polynomials over Finite Fields

2003 ◽  
Vol 55 (2) ◽  
pp. 225-246 ◽  
Author(s):  
William D. Banks ◽  
Asma Harcharras ◽  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Small Degree ◽  
Kloosterman Sums ◽  
Degree Relative ◽  
Irreducible Polynomials ◽  
Arbitrary Polynomial ◽  
Polynomials Over Finite Fields ◽  
Titchmarsh Theorem

AbstractWe extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring [x]/M(x) for collections of polynomials either of the form f−1g−1 or of the form f−1g−1 + afg, where f and g are polynomials coprime to M and of very small degree relative to M, and a is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields.

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