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

Todd Cochrane, et al. "Waring's number for large subgroups of ℤ*ₚ*." Acta Arithmetica 163.4 (2014): 309-325. <http://eudml.org/doc/286595>.

@article{ToddCochrane2014,
abstract = {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 $t₄ = p^\{22/39+ϵ\}$, $t₅ = p^\{15/29+ϵ\}$ and for s ≥ 6, $tₛ = p^\{(9s+45)/(29s+33)+ϵ\}$. For s ≥ 24 further improvements are made, such as $t_\{32\} = p^\{5/16+ϵ\}$ and $t_\{128\} = p^\{1/4\}$.},
author = {Todd Cochrane, Derrick Hart, Christopher Pinner, Craig Spencer},
journal = {Acta Arithmetica},
keywords = {Waring's problem; exponential sums; sum-product sets},
language = {eng},
number = {4},
pages = {309-325},
title = {Waring's number for large subgroups of ℤ*ₚ*},
url = {http://eudml.org/doc/286595},
volume = {163},
year = {2014},
}

TY - JOUR
AU - Todd Cochrane
AU - Derrick Hart
AU - Christopher Pinner
AU - Craig Spencer
TI - Waring's number for large subgroups of ℤ*ₚ*
JO - Acta Arithmetica
PY - 2014
VL - 163
IS - 4
SP - 309
EP - 325
AB - 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 $t₄ = p^{22/39+ϵ}$, $t₅ = p^{15/29+ϵ}$ and for s ≥ 6, $tₛ = p^{(9s+45)/(29s+33)+ϵ}$. For s ≥ 24 further improvements are made, such as $t_{32} = p^{5/16+ϵ}$ and $t_{128} = p^{1/4}$.
LA - eng
KW - Waring's problem; exponential sums; sum-product sets
UR - http://eudml.org/doc/286595
ER -