On the representation of polynomials over finite fields as sums of powers and irreducibles
William A. Webb · Rocky Mountain Journal of Mathematics · 1973
I. Introduction.There are a number of results known concerning the expression of an integer as the sum of a certain number of primes and fcth powers [2], [3], [4].In this paper, we prove several of these results, specifically those found in [4], for polynomial rings over finite fields.A Hardy-Littlewood like method is used.The use of the Riemann hypothesis simplifies die proofs and enables us to obtain better error terms than those obtained in [4]. II. Notation and preliminary results.In general we follow the notation used in [5] and [6].GF [q,x] is the ring of polynomials over the finite field with q elements, q = p ß , p a prime.D( i/ x is the completion of the field of rational functions over GF(q), with respect to *>, the degree valuation. j}. k, and N^K) is the number of representations of K in the form