EUDML

For a prime $p$ and positive integers $ℓ < k < h < p$ with $d = (h, k, ℓ, p - 1)$ , we show that $M$ , the number of simultaneous solutions $x, y, z, w$ in $ℤ_{p}^{*}$ to $x^{h} + y^{h} = z^{h} + w^{h}$ , $x^{k} + y^{k} = z^{k} + w^{k}$ , $x^{ℓ} + y^{ℓ} = z^{ℓ} + w^{ℓ}$ , satisfies $M \leq 3 d^{2} {(p - 1)}^{2} + 25 h k ℓ (p - 1) .$ When $h k ℓ = o (p d^{2})$ we obtain a precise asymptotic count on $M$ . This leads to the new twisted exponential sum bound $|\sum_{x = 1}^{p - 1} χ (x) e^{2 π i f (x) / p}| \leq 3^{\frac{1}{4}} d^{\frac{1}{2}} p^{\frac{7}{8}} + \sqrt{5} {(h k ℓ)}^{\frac{1}{4}} p^{\frac{5}{8}},$ for trinomials $f = a x^{h} + b x^{k} + c x^{ℓ}$ , and to results on the average size of such sums.

Sparse polynomial exponential sums

Todd Cochrane; Christopher Pinner; Jason Rosenhouse — 2003

Acta Arithmetica

A further refinement of Mordell's bound on exponential sums

Todd Cochrane; Jeremy Coffelt; Christopher Pinner — 2005

Acta Arithmetica

On the parity of k-th powers modulo p. A generalization of a problem of Lehmer

Jean Bourgain; Todd Cochrane; Jennifer Paulhus; Christopher Pinner — 2011

Acta Arithmetica

Waring's number for large subgroups of ℤₚ

Todd Cochrane; Derrick Hart; Christopher Pinner; Craig Spencer — 2014

Acta Arithmetica

Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain $t ₄ = p^{22 / 39 + ϵ}$ , $t ₅ = p^{15 / 29 + ϵ}$ and for s ≥ 6, $t ₛ = p^{(9 s + 45) / (29 s + 33) + ϵ}$ . For s ≥ 24 further improvements are made, such as $t_{32} = p^{5 / 16 + ϵ}$ and $t_{128} = p^{1 / 4}$ .