The factorable core of polynomials over finite fields

Abstract For a polynomial f(x) over a finite field q , denote F the polynomial f{y)-f(x) by f(x,y).The polynomial ipj has frequently been used in questions on the values of /. The existenceis proved here of a polynomial F ove q of thr Fe form F r , = wher L e L is an affine lin-earized polynomial over F , such that / = g(F) for some polynomial g q>r and the part ofwhich splits completely into linear factors over the algebraic closur q ie osf exactl ¥ <py F . Thisilluminates an aspect of work of D. R. Hayes and Daqing Wan on the existence of permutationpolynomials of even degree. Related results on value sets, including the exhibition of a class ofpermutation polynomials, are also mentioned.1980 Mathematics subject classification (Anter. Math. Soc.) (1985 Revision): 11 T 06. 1. Introduction Let f(x) be a monic polynomia [x]l i , whern Fe F is the finite field ofprime power order q = p m . (Without loss, we shall assume throughout that/ is separable, that is, f(x) $ F^^].) Questions relating to the value setof / in

[1]  Stephen D. Cohen,et al.  UNIFORM DISTRIBUTION OF POLYNOMIALS OVER FINITE FIELDS , 1972 .

[2]  Wang Daqing.  On a Conjecture of Carlitz , 1987, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[3]  Daniel J. Madden,et al.  Polynomials with small value set over finite fields , 1988 .

[4]  Stephen D. Cohen.  REDUCIBILITY OF SUB-LINEAR POLYNOMIALS OVER A FINITE FIELD , 1985 .

[5]  Leonard Carlitz.  On factorable polynomials in several indeterminates , 1936 .

[6]  L. Dickson.  The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group. , 1896 .

[7]  M. Fried,et al.  On curves with separated variables , 1969 .

[8]  Sur la factorisation des polynomes $$X^{p^{2r} + p^r + 1} - aX^{p^r + 1} - bX - c$$ sur les corps finis $$IF_{s_p } $$ , 1987 .

[9]  Stephen D. Cohen,et al.  The distribution of polynomials over finite fields , 1970 .

[10]  A note on polynomials with minimal value set over finite fields , 1988 .

[11]  The degrees of the factors of certain polynomials over finite fields. , 1970 .

[12]  David R. Hayes.  A geometric approach to permutation polynomials over a finite field , 1967 .

[13]  R. Lidl,et al.  When does a polynomial over a finite field permute the elements of the fields , 1988 .

[14]  William Mills.  Polynomials with minimal value sets , 1964 .

[15]  L. Carlitz.  On the number of distinct values of a polynomial with coefficients in a finite field , 1955 .

[16]  Rudolf Lide,et al.  Finite fields , 1983 .