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

• Autorzy: Todd Cochrane , Derrick Hart , Christopher Pinner , Craig Spencer
• Języki publikacji: EN

Abstrakty

EN

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 $t_{128} = p^{1/4}$.

Kategorie tematyczne

• 11L07: Estimates on exponential sums
• 11P05: Waring's problem and variants
• 11B30: Arithmetic combinatorics; higher degree uniformity

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 163
• Numer: 4
• Strony: 309-325

Daty

wydano

2014

Twórcy

autor

Todd Cochrane

• Department of Mathematics, Kansas State University, Manhattan, KS 66506, U.S.A.

autor

Derrick Hart

• Department of Mathematics, Rockhurst University, Kansas City, MO 64110, U.S.A.

autor

Christopher Pinner

• Department of Mathematics, Kansas State University, Manhattan, KS 66506, U.S.A.

autor

Craig Spencer

• Department of Mathematics, Kansas State University, Manhattan, KS 66506, U.S.A.

Identyfikatory

DOI

10.4064/aa163-4-2