Waring’s number in finite fields

Abstract

This thesis establishes bounds (primarily upper bounds) on Waring's number in finite fields. Let p be a prime, q=pn, Fq be the finite field in q elements, k be a positive integer with k|(q−1) and t=(q−1)/k. Let Ak denote the set of k-th powers in Fq and Ak′=Ak∩Fp. Waring's number γ(k,q) is the smallest positive integer s such that every element of Fq can be expressed as a sum of s k-th powers. For prime fields Fp we prove that for any positive integer r there is a constant C(r) such that γ(k,p)≤C(r)k1/r provided that ϕ(t)≥r. We also obtain the lower bound γ(k,p)≥(t−1)ek1/(t−1)−t+1 for t prime. For general finite fields we establish the following upper bounds whenever γ(k,q) exists: γ(k,q)≤7.3n⌈(2k)1/n|Ak′|−1⌉log⁡(k),γ(k,q)≤8n⌈(k+1)1/n−1|Ak′|−1⌉,and γ(k,q)≪n(k+1)log⁡(4)nlog⁡|\kp|log⁡log⁡(k).We also establish the following versions of the Heilbronn conjectures for general finite fields. For any \ve>0 there is a constant, c(\ve), such that if |Ak′|≥42\ven, then γ(k,q)≤c(ε)kε. Next, if n≥3 and γ(k,q) exists, then γ(k,q)≤10k+1. For n=2, we have γ(k,p2)≤16k+1.