A cyclotomic approach to the solution of Waring's problem mod p

Abstract

Let $s_d(p,a) = \min \{k | a = \sum_{i=1}^{k}a_i^d, a_i\in \ff_p^*\}$ be the smallest number of d-th powers in the finite field F_p, sufficient to represent the number a in F_p^*. Then $$g_d(p) = max_{a in F_p^*} s_d(p,a)$$ gives an answer to Waring's Problem mod p. We first introduce cyclotomic integers n(k,\nu), which then allow to state and solve Waring's problem mod p in terms of only the cyclotomic numbers (i,j) of order d. We generalize the reciprocal of the Gaussian period equation G(T) to a C-differentiable function I(T) in Q[[T]], which also satisfies I'(T)/I(T) in Z[[T]]. We show that and why a\equiv -1 mod F_p^{*d} (the classical "Stufe", if d = 2) behaves special: Here (and only here) I(T) is in fact a polynomial from Z[T], the reciprocal of the period polynomial. We finish with explicit calculations of g_d(p) for the cases d = 3 and d = 4, all primes p, using the known cyclotomic numbers compiled by Dickson.