The factorable core of polynomials over finite fields

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.

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SOME FAMILIES OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS

International Journal of Number Theory ◽

10.1142/s1793042108001717 ◽

2008 ◽

Vol 04 (05) ◽

pp. 851-857 ◽

Cited By ~ 24

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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A link between minimal value set polynomials and tamely ramified towers of function fields over finite fields

Journal of Algebra and Its Applications ◽

10.1142/s0219498822501894 ◽

2021 ◽

pp. 2250189

Author(s):

R. Toledano

Keyword(s):

Finite Field ◽

Finite Fields ◽

Finite Subset ◽

Function Fields ◽

Algebraic Closure ◽

Value Set ◽

Towers Of Function Fields ◽

Value Sets

In this paper, we introduce the notions of [Formula: see text]-polynomial and [Formula: see text]-minimal value set polynomial where [Formula: see text] is a polynomial over a finite field [Formula: see text] and [Formula: see text] is a finite subset of an algebraic closure of [Formula: see text]. We study some properties of these polynomials and we prove that the polynomials used by Garcia, Stichtenoth and Thomas in their work on good recursive tame towers are [Formula: see text]-minimal value set polynomials for [Formula: see text], whose [Formula: see text]-value sets can be explicitly computed in terms of the monomial [Formula: see text].

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On some permutation polynomials over finite fields

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.2631 ◽

2005 ◽

Vol 2005 (16) ◽

pp. 2631-2640 ◽

Cited By ~ 8

Author(s):

Amir Akbary ◽

Qiang Wang

Keyword(s):

Finite Field ◽

Recurrence Relation ◽

Finite Fields ◽

Permutation Polynomials ◽

Initial Values ◽

Polynomials Over Finite Fields

Letpbe prime,q=pm, andq−1=7s. We completely describe the permutation behavior of the binomialP(x)=xr(1+xes)(1≤e≤6) over a finite fieldFqin terms of the sequence{an}defined by the recurrence relationan=an−1+2an−2−an−3(n≥3) with initial valuesa0=3,a1=1, anda2=5.

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Dickson Polynomials Over Finite Fields and Complete Mappings

Canadian Mathematical Bulletin ◽

10.4153/cmb-1987-003-3 ◽

1987 ◽

Vol 30 (1) ◽

pp. 19-27 ◽

Cited By ~ 21

Author(s):

Gary L. Mullen ◽

Harald Niederreiter

Keyword(s):

Finite Field ◽

Finite Fields ◽

Permutation Polynomials ◽

Dickson Polynomials ◽

Complete Mapping ◽

Dickson Polynomial ◽

Special Cases ◽

Polynomials Over Finite Fields ◽

Complete Mappings ◽

Special Case

AbstractDickson polynomials over finite fields are familiar examples of permutation polynomials, i.e. of polynomials for which the corresponding polynomial mapping is a permutation of the finite field. We prove that a Dickson polynomial can be a complete mapping polynomial only in some special cases. Complete mapping polynomials are of interest in combinatorics and are defined as polynomials f(x) over a finite field for which both f(x) and f(x) + x are permutation polynomials. Our result also verifies a special case of a conjecture of Chowla and Zassenhaus on permutation polynomials.

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On permutation polynomials over finite fields

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171287000644 ◽

1987 ◽

Vol 10 (3) ◽

pp. 535-543 ◽

Cited By ~ 13

Author(s):

R. A. Mollin ◽

C. Small

Keyword(s):

Finite Field ◽

Finite Fields ◽

Permutation Polynomial ◽

Permutation Polynomials ◽

Polynomials Over Finite Fields ◽

One To One ◽

Necessary And Sufficient

A polynomialfover a finite fieldFis called a permutation polynomial if the mappingF→Fdefined byfis one-to-one. In this paper we consider the problem of characterizing permutation polynomials; that is, we seek conditions on the coefficients of a polynomial which are necessary and sufficient for it to represent a permutation. We also give some results bearing on a conjecture of Carlitz which says essentially that for any even integerm, the cardinality of finite fields admitting permutation polynomials of degreemis bounded.

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Constructions and Necessities of Some Permutation Polynomials over Finite Fields

Mathematical Problems in Engineering ◽

10.1155/2021/9409130 ◽

2021 ◽

Vol 2021 ◽

pp. 1-13

Author(s):

Xiaogang Liu

Keyword(s):

Finite Field ◽

Finite Fields ◽

Coding Theory ◽

Combinatorial Design ◽

Permutation Polynomials ◽

Science And Engineering ◽

Polynomials Over Finite Fields

Let F q denote the finite field with q elements. Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, and combinatorial design. The study of permutation polynomials has a long history, and many results are obtained in recent years. In this paper, we obtain some further results about the permutation properties of permutation polynomials. Some new classes of permutation polynomials are constructed, and the necessities of some permutation polynomials are studied.

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On permutation polynomials over finite fields of characteristic 2

Journal of Algebra and Its Applications ◽

10.1142/s0219498816501334 ◽

2016 ◽

Vol 15 (07) ◽

pp. 1650133 ◽

Cited By ~ 1

Author(s):

Rohit Gupta ◽

R. K. Sharma

Keyword(s):

Finite Field ◽

Finite Fields ◽

Permutation Polynomial ◽

Permutation Polynomials ◽

Polynomials Over Finite Fields ◽

Characteristic 2 ◽

Polynomial Formula

Let [Formula: see text] denotes the finite field of order [Formula: see text] where [Formula: see text] A permutation polynomial [Formula: see text] over [Formula: see text] with [Formula: see text] and [Formula: see text] such that for each [Formula: see text] is a permutation polynomial satisfying [Formula: see text] is called a o-polynomial. In this paper, we determine all o-polynomials up to degree [Formula: see text].

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