Carlitz [Solvability of certain equations in a finite field, Quart. J. Math. (Oxford) 7 (1956), 3–4] determined conditions under which infinite families of polynomials have solutions in a finite field. In this paper we extend some of Carlitz's results by computing the exact p-divisibility of certain exponential sums. As a by-product we obtain an upper bound for the Waring number for polynomials over extensions of finite fields.



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