A class of permutation trinomials over finite fields

Let Fq denote the finite field with q elements. A polynomial f ∈ Fq[x] is called a permutation polynomial (PP) of Fq if the mapping x 7→ f(x) is a permutation of Fq. There is always great interest in permutation polynomials that appear in simple algebraic forms. To this end, a considerable amount of research has been devoted to finding and understanding permutation binomials. For a few samples from a long list of publications on permutation binomials over finite fields, see [1, 4, 9, 13, 15, 16, 17, 18, 19, 20]. As for permutation trinomials, there are not many theoretic results. Discoveries of infinite classes of permutation trinomials are less frequent than those of permutation binomials [3, 6, 14]. The main results of the present paper are the following theorems.

[1]  Qiang Wang,et al.  The Number of Permutation Binomials over F4p+1 where p and 4p+1 are Primes , 2006, Electron. J. Comb..

[2]  June-Bok Lee,et al.  SOME PERMUTING TRINOMIALS OVER FINITE FIELDS , 1997 .

[3]  Doron Zeilberger,et al.  An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities , 1992 .

[4]  Xiang-dong Hou,et al.  A Class of Permutation Binomials over Finite Fields , 2012, 1210.0881.

[5]  Qiang Wang,et al.  A generalized Lucas sequence and permutation binomials , 2005 .

[6]  L. Carlitz,et al.  Some theorems on permutation polynomials , 1962 .

[7]  Doron Zeilberger,et al.  Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory , 2006, Adv. Appl. Math..

[8]  R. Tennant.  Algebra , 1941, Nature.

[9]  Michael E. Zieve.  On some permutation polynomials over Fq of the form x^r h(x^{(q-1)/d}) , 2007, 0707.1110.

[10]  Xiang-dong Hou,et al.  Reversed Dickson polynomials over finite fields , 2009, Finite Fields Their Appl..

[11]  Ariane M. Masuda,et al.  Permutation binomials over finite fields , 2007, 0707.1108.

[12]  Neranga Fernando,et al.  A new approach to permutation polynomials over finite fields, II , 2012, Finite Fields Their Appl..

[13]  Wang Daqing,et al.  Permutation polynomials over finite fields , 1987 .

[14]  Lei Hu,et al.  Two classes of permutation polynomials over finite fields , 2012, Finite Fields Their Appl..

[15]  Irving Kaplansky,et al.  Fields and rings , 1969 .

[16]  N. Vasilev,et al.  Permutation binomials and their groups , 2011 .