FINITE FIELDS AND THEIR APPLICATIONS 2, 337–347 (1996) ARTICLE NO. 0021 On the Value Sets of Special Polynomials over Finite Fields ABRAMO HEFEZ* Departamento de Matema ´tica Aplicada, Universidade Federal Fluminense, R. Sa ˜o Paulo s/n–Campus do Valoguinho, 24020-005 Nitero ´ i, RJ, Brazil Communicated by Gerhard Turnwald Received October 17, 1994; revised November 8, 1995 1. INTRODUCTION Let V f = f ( q ) be the value set of a polynomial f (x)   q [x], where  q is a finite field with q elements of characteristic p. A fundamental problem is to estimate the size of V f , and a powerful strategy often used to determine lower bounds for the cardinality |V f | of V f is to estimate, usually by means of Weil’s bound, the number of solutions N f* in  q   q of the equation f *(x, y):= f (x) - f ( y) = 0, and use Uchiyama’s inequality (see [2, Lemma 2; 9; or 10, Lemma 1]), |V f |  q 2 N f * . In this paper we will study the value sets of the special polynomials of the form f (x) = (x m + b) n , * The author was partially supported by CNPq-Brazil. 337 1071-5797/96 $18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.