ON SEMIGROUP ORBITS OF POLYNOMIALS AND MULTIPLICATIVE ORDERS

We show, under some natural restrictions, that some semigroup orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime $p$, extending previous work of Shparlinski [‘Multiplicative orders in orbits of polynomials over finite fields’, Glasg. Math. J.60(2) (2018), 487–493].

MULTIPLICATIVE ORDERS IN ORBITS OF POLYNOMIALS OVER FINITE FIELDS

Glasgow Mathematical Journal ◽

10.1017/s0017089517000222 ◽

2017 ◽

Vol 60 (2) ◽

pp. 487-493 ◽

Author(s):

IGOR E. SHPARLINSKI

Keyword(s):

Finite Fields ◽

Algebraic Closure ◽

Rational Numbers ◽

Duke Math ◽

Roots Of Unity ◽

Multiplicative Order ◽

Polynomial Orbits

AbstractWe show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p. We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p) is large. The approach is based on the results of Dvornicich and Zannier (Duke Math. J.139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers.

On the number of factorizations of polynomials over finite fields

Journal of Combinatorial Theory Series A ◽

10.1016/j.jcta.2021.105462 ◽

2021 ◽

Vol 182 ◽

pp. 105462

Author(s):

Rachel N. Berman ◽

Ron M. Roth

Keyword(s):

Finite Fields ◽

Some primitive polynomials over finite fields

Acta Mathematica Scientia ◽

10.1016/s0252-9602(17)30428-9 ◽

2001 ◽

Vol 21 (3) ◽

pp. 412-416 ◽

Author(s):

Seunghwan Chang ◽

June Bok Lee

Keyword(s):

Finite Fields ◽

Primitive Polynomials ◽

Factorization of a class of polynomials over finite fields

Finite Fields and Their Applications ◽

10.1016/j.ffa.2011.07.005 ◽

2012 ◽

Vol 18 (1) ◽

pp. 108-122 ◽

Cited By ~ 17

Author(s):

Henning Stichtenoth ◽

Alev Topuzoğlu

Keyword(s):

Finite Fields ◽

Value sets of polynomials over finite fields

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1993-1155603-2 ◽

1993 ◽

Vol 119 (3) ◽

pp. 711-711 ◽

Cited By ~ 15

Author(s):

Da Qing Wan ◽

Peter Jau-Shyong Shiue ◽

Ching Shyang Chen

Keyword(s):

Finite Fields ◽

Value Sets

On a conjecture of Fernando, Hou and Lappano concerning permutation polynomials over finite fields

Finite Fields and Their Applications ◽

10.1016/j.ffa.2018.11.006 ◽

2019 ◽

Vol 56 ◽

pp. 58-92 ◽

Author(s):

Wun-Seng Chou ◽

Xiang-dong Hou

Keyword(s):

Finite Fields ◽

Permutation Polynomials ◽

Construction of authentication codes with distrust arbitration from polynomials over finite fields

The Journal of China Universities of Posts and Telecommunications ◽

10.1016/s1005-8885(17)60191-2 ◽

2017 ◽

Vol 24 (1) ◽

pp. 87-95

Author(s):

Chen Shangdi ◽

Tian Wenjing ◽

Li Xue

Keyword(s):

Finite Fields ◽

Authentication Codes ◽

The degrees of permutation polynomials over finite fields

Journal of Combinatorial Theory ◽

10.1016/s0021-9800(69)80005-0 ◽

1969 ◽

Vol 7 (1) ◽

pp. 49-55 ◽

Cited By ~ 10

Author(s):

Charles Wells

Keyword(s):

Finite Fields ◽

Permutation Polynomials ◽

The Kluwer International Series in Engineering and Computer Science - Finite Fields for Computer Scientists and Engineers ◽

Factoring Polynomials over Finite Fields

10.1007/978-1-4613-1983-2_7 ◽

1987 ◽

pp. 75-96 ◽

Cited By ~ 2

Author(s):

Robert J. McEliece

Keyword(s):

Finite Fields ◽

Factoring Polynomials

Short Kloosterman Sums for Polynomials over Finite Fields

Canadian Journal of Mathematics ◽

10.4153/cjm-2003-010-0 ◽

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 ◽

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.