Groups generated by iterations of polynomials over finite fields

2015 ◽  
Vol 59 (1) ◽  
pp. 235-245 ◽  
Author(s):  
Igor E. Shparlinski
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Finite Fields ◽  
Mild Condition ◽  
Character Sums ◽  
Small Subgroup ◽  
Polynomials Over Finite Fields

AbstractGiven a finite field of q elements, we consider a trajectory of the map associated with a polynomial ]. Using bounds of character sums, under some mild condition on f, we show that for an appropriate constant C > 0 no N ⩾ Cq½ distinct consecutive elements of such a trajectory are contained in a small subgroup of , improving the trivial lower bound . Using a different technique, we also obtain a similar result for very small values of N. These results are multiplicative analogues of several recently obtained bounds on the length of intervals containing N distinct consecutive elements of such a trajectory.

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.

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.

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.

A SPECIAL DEGREE REDUCTION OF POLYNOMIALS OVER FINITE FIELDS WITH APPLICATIONS

2011 ◽  
Vol 07 (04) ◽  
pp. 1093-1102 ◽  
Author(s):  
WEI CAO
Keyword(s):  
Finite Field ◽  
Explicit Formula ◽  
Finite Fields ◽  
Rational Points ◽  
Degree Reduction ◽  
Affine Hypersurface ◽  
Special Degree ◽  
Low Degree ◽  
Polynomials Over Finite Fields ◽  
Degree Matrix

Let f be a polynomial in n variables over the finite field 𝔽q and Nq(f) denote the number of 𝔽q-rational points on the affine hypersurface f = 0 in 𝔸n(𝔽q). A φ-reduction of f is defined to be a transformation σ : 𝔽q[x1, …, xn] → 𝔽q[x1, …, xn] such that Nq(f) = Nq(σ(f)) and deg f ≥ deg σ(f). In this paper, we investigate φ-reduction by using the degree matrix which is formed by the exponents of the variables of f. With φ-reduction, we may improve various estimates on Nq(f) and utilize the known results for polynomials with low degree. Furthermore, it can be used to find the explicit formula for Nq(f).

scholarly journals The reducibility theorem for linearised polynomials over finite fields

1989 ◽  
Vol 40 (3) ◽  
pp. 407-412 ◽  
Author(s):  
Stephen D. Cohen
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Irreducible Polynomials ◽  
Polynomials Over Finite Fields ◽  
Wide Readership

A self-contained elementary account is given of the theorem of S. Agou that classifies all composite irreducible polynomials of the form over a finite field of characteristic p. Written to appeal to a wide readership, it is intended to complement the original rather technical proof and other contributions by the author and by Moreno.

scholarly journals The factorable core of polynomials over finite fields

1990 ◽  
Vol 49 (2) ◽  
pp. 309-318 ◽  
Author(s):  
S. D. Cohen
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Algebraic Closure ◽  
Permutation Polynomials ◽  
Linearized Polynomial ◽  
Polynomials Over Finite Fields ◽  
Value Sets

AbstractFor a polynomial f(x) over a finite field Fq, denote the polynomial f(y)−f(x) by ϕf(x, y). The polynomial ϕf has frequently been used in questions on the values of f. The existence is proved here of a polynomial F over Fq of the form F = Lr, where L is an affine linearized polynomial over Fq, such that f = g(F) for some polynomial g and the part of ϕf which splits completely into linear factors over the algebraic closure of Fq is exactly φF. This illuminates an aspect of work of D. R. Hayes and Daqing Wan on the existence of permutation polynomials of even degree. Related results on value sets, including the exhibition of a class of permutation polynomials, are also mentioned.

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

scholarly journals SOME FAMILIES OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS

2008 ◽  
Vol 04 (05) ◽  
pp. 851-857 ◽  
Author(s):  
MICHAEL E. ZIEVE
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Permutation Polynomials ◽  
Polynomials Over Finite Fields ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for a polynomial of the form xr(1 + xv + x2v + ⋯ + xkv)t to permute the elements of the finite field 𝔽q. Our results yield especially simple criteria in case (q - 1)/ gcd (q - 1, v) is a small prime.

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