Polynomials with small value set over finite fields

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 [{norm of matrix}] 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 d⁴ < 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 | (q² - 1), 3. (c) f(x) = ((x + a)² + b) ^{d 2} + c, where d | (q² - 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 | (q² - 1) and a is a 2^kth power in K_q² where d = 2^kr, 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.