On the value sets of special polynomials over finite fields (Q1383507)

The author studies the value set over the finite field \({\mathbb{F}}_q\) of a polynomial of the form \((x^m+b)^n\), where \(b\in {\mathbb{F}}_q^*\) and \(m\) and \(n\) are integers greater than or equal to 2 that divide \(q-1\). The value set of such a polynomial was studied previously by \textit{J. Gomez-Calderon} [Rocky Mt. J. Math. 23, No. 1, 111-118 (1993; Zbl 0778.11070)], who obtained a lower bound for the cardinality of such a value set if \(mn<q^{1/4}\). Part of the argument used by Gomez-Calderon involved applying Weil's bound to bound the number of rational points on a certain Fermat curve. The author obtains new lower bounds on the cardinality of such a value set by applying improved bounds on the number of rational points on a Fermat curve that have been shown by \textit{A. Garcia} and \textit{J. F. Voloch} [J. Number Theory 30, 345-356 (1988; Zbl 0671.14012)], \textit{A. Hefez} and \textit{J. F. Voloch} [Arch. Math. 54, 263-273 (1990; Zbl 0662.14016)], and the author and \textit{N. Kakuta} [Proc. Zeuthen Symp., Copenhagen/Den. 1989, Contemp. Math. 123, 89-97 (1991; Zbl 0749.14023)].