Exact solutions to Waring's problem for finite fields
Abstract
The Waring function g(k,q) measures the difficulty of Waring's problem for kth powers in the field of q elements. Its calculation seems to be difficult, and many partial results have been published, notably upper bounds for certain regions of the k-q-plane. In this paper, we compute the exact value of g(k,q) for two infinite families of exponent-field pairs. In these, k is large compared to q. We use a new method of proof that is mainly combinatorial in nature.
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