Currently displaying 1 – 12 of 12

Showing per page

Order by Relevance | Title | Year of publication

A system of simultaneous congruences arising from trinomial exponential sums

Todd CochraneJeremy CoffeltChristopher Pinner — 2006

Journal de Théorie des Nombres de Bordeaux

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

Waring's number for large subgroups of ℤ*ₚ*

Todd CochraneDerrick HartChristopher PinnerCraig 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 = p 22 / 39 + ϵ , t = p 15 / 29 + ϵ and for s ≥ 6, t = 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 .

Page 1

Download Results (CSV)