Polynomials with small value set over finite fields

Abstract

Let K q denote the finite field with q elements and characteristic p. Let f( x) be a monic polynomial of degree d with coeficients in K q . Let C( f) denote the number of distinct values of f( x) as x ranges over K q . We easily see that C(f)⩾ q−1 d +1 where [‖] is the greatest integer function. A polynomial for which equality in (∗) occurs is called a minimum value set polynomial. There is a complete characterization of minimum value set polynomials over arbitrary finite fields with d < √ q and C( f) ≥ 3 [see L. Carlitz, D. J. Lewis, W. H. Mills, and E. G. Strauss, Mathematika 8 (1961), 121–130; W. H. Mills, Pacific J. Math. 14 (1964), 225–241]. In this paper we give a complete list of polynomials of degree d 4 < q which have a value set of size less than 2q d , twice the minimum possible. If d > 15 then f( x) is one of the following polynomial forms: 1. (a) f( x) = ( x + a) d + b, where d | ( q − 1), 2. (b) f(x) = ((x + a) d 2 + b) 2 + c , where d | ( q 2 − 1), 3. (c) f(x) = ((x + a) 2 + b) d 2 + c , where d | ( q 2 − 1), or 4. (d) f( x) = D d, a ( x + b) + c, where D d, a ( x) is the Dickson polynomial of degree d, d | ( q 2 − 1) and a is a 2 k th power in K q 2 where d = 2 k r, r is odd. The result is obtained by noticing the connection between the size of the value set of a polynomial f( x) and the factorization of the associated substitution polynomial f ∗(x, y) = f(x) − f(y) in the ring K q [ x, y]. Essentually, we show that C(f) < 2q d implies that f ∗(x, y) has at least d 2 factors in K q [ x, y], and we determine all the polynomials with such characteristic.