scholarly journals ARITHMETIC AND GEOMETRIC PROGRESSIONS IN PRODUCT SETS OVER FINITE FIELDS

2008 ◽  
Vol 78 (3) ◽  
pp. 357-364
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Arithmetic Progression ◽  
Geometric Progressions ◽  
Image Position ◽  
Product Sets

AbstractGiven two sets ${\mathcal A}, {\mathcal B} \subseteq \mathbb {F}_q$ of elements of the finite field 𝔽q of q elements, we show that the product set contains an arithmetic progression of length k≥3 provided that k<p, where p is the characteristic of 𝔽q, and #𝒜#ℬ≥3q2d−2/k. We also consider geometric progressions in a shifted product set 𝒜ℬ+h, for f∈𝔽q, and obtain a similar result.

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.

Zeros of random polynomials over finite fields

1992 ◽  
Vol 111 (2) ◽  
pp. 193-197 ◽  
Author(s):  
R. W. K. Odoni
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Prime Power ◽  
Random Polynomials ◽  
Polynomials Over Finite Fields ◽  
Image Position

Let be the finite field with q elements (q a prime power), let r 1 and let X1, , Xr be independent indeterminates over . We choose an arbitrary and a d 1 and consider

Diagonal Equations Over Large Finite Fields

1984 ◽  
Vol 36 (2) ◽  
pp. 249-262 ◽  
Author(s):  
Charles Small
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Finite Extension ◽  
Extension Field ◽  
Explicit Bound ◽  
Image Position

We consider polynomials of the formwith non-zero coefficients ai in a finite field F. For any finite extension field K ⊇ F, let fk:Kn → K be the mapping defined by f. We say f is universal over K if fK is surjective, and f is isotropic over K if fK has a non-trivial “kernel“; the latter means fK(X) = 0 for some 0 ≠ x ∊ Kn.We show (Theorem 1) that f is universal over K provided |K| (the cardinality of K) is larger than a certain explicit bound given in terms of the exponents d1,…, dn. The analogous fact for isotropy is Theorem 2.It should be noted that in studying diagonal equationswe fix both the number of variables n and the exponents di, and ask how large the field must be to guarantee a solution.

scholarly journals ON THE SOLVABILITY OF BILINEAR EQUATIONS IN FINITE FIELDS

2008 ◽  
Vol 50 (3) ◽  
pp. 523-529 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Absolute Constant ◽  
Character Sums ◽  
Multiplicative Character ◽  
Image Position ◽  
Multiplicative Character Sums ◽  
Bilinear Equations

AbstractWe consider the equation over a finite field q of q elements, with variables from arbitrary sets $\cA,\cB, \cC, \cD \subseteq \F_q$. The question of solvability of such and more general equations has recently been considered by Hart and Iosevich, who, in particular, prove that if for some absolute constant C > 0, then above equation has a solution for any λ ∈ q*. Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property.

scholarly journals Further Arithmetical Functions in Finite Fields

1969 ◽  
Vol 16 (4) ◽  
pp. 349-363 ◽  
Author(s):  
Stephen D. Cohen
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Ordered Set ◽  
Equivalence Classes ◽  
Irreducible Polynomials ◽  
Arithmetical Functions ◽  
Image Position

In this paper, the author continues his investigation, initiated in (4) and (5), into the nature of certain “arithmetical” functions associated with the factorisation of normalised non-zero polynomials in the ring GF[q, X1, …, Xk], where k ≧ 1, GF(q) is the finite field of order q and X1, …, Xk are indeterminates. By normalised polynomials we mean that exactly one polynomial has been selected from equivalence classes with respect to multiplication by non-zero elements of GF(q). With this normalisation GF[q, X1, …, Xk] becomes a unique factorisation domain. The constant polynomial will be denoted by 1. By the degree of a polynomial A in GF[q, X1, …, Xk], we shall mean the ordered set (m1, …, mk), where mi is the degree of A in Xi, 1 ≦ i ≦.k. We shall assume that A(≠ 1), a typical polynomial in GF[q, X1, … Xk], has prime factorisationwhere P1, …, Pr are distinct irreducible polynomials (i.e. primes).

On irreducible polynomials of certain types in finite fields

1969 ◽  
Vol 66 (2) ◽  
pp. 335-344 ◽  
Author(s):  
Stephen D. Cohen
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Monic Polynomial ◽  
Irreducible Polynomials ◽  
Image Position

Let GF(q) be the finite field containing q = pl elements, where p is a prime and l a positive integer. Let P(x) be a monic polynomial in GF[q, x] of degree m. In this paper we investigate the nature and distribution of monic irreducible polynomials of the following types:(I) P(xr), where r is a positive integer (r-polynomials).(II) xm P(x + x−1). (Reciprocal polynomials.) These have the form(III) xrmP(xr + x−r). (r-reciprocal polynomials.) These have the form Q(xr), where q(x) satisfies (1·1).

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.

scholarly journals Monomial Generalized Almost Perfect Nonlinear Functions

2020 ◽  
Vol 31 (03) ◽  
pp. 411-419
Author(s):  
Masamichi Kuroda
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Algebraic Degree ◽  
Apn Functions ◽  
Nonlinear Functions ◽  
Almost Perfect Nonlinear ◽  
Basic Properties ◽  
Almost Perfect Nonlinear Functions

Generalized almost perfect nonlinear (GAPN) functions were defined to satisfy some generalizations of basic properties of almost perfect nonlinear (APN) functions for even characteristic. In particular, on finite fields of even characteristic, GAPN functions coincide with APN functions. In this paper, we study monomial GAPN functions for odd characteristic. We give monomial GAPN functions whose algebraic degree are maximum or minimum on a finite field of odd characteristic. Moreover, we define a generalization of exceptional APN functions and give typical examples.

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