Functiones et Approximatio Commentarii Mathematici

The unrestricted variant of Waring's problem in function fields

Yu-Ru Liu and Trevor D. Wooley

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Abstract

Let $\mathbb{J}_q^k[t]$ denote the additive closure of the set of $k$th powers in the polynomial ring $\mathbb{F}_q[t]$, defined over the finite field $\mathbb{F}_q$ having $q$ elements. We show that when $s\ge k+1$ and $q \ge k^{2k+2}$, then every polynomial in $\mathbb{J}_q^k[t]$ is the sum of at most $s$ $k$th powers of polynomials from $\mathbb{F}_q[t]$. When $k$ is large and $s \ge (\frac{4}{3}+o(1)) k\log k$, the same conclusion holds without restriction on $q$. Refinements are offered that depend on the characteristic of $\mathbb{F}_q$.

Article information

Source
Funct. Approx. Comment. Math., Volume 37, Number 2 (2007), 285-291.

Dates
First available in Project Euclid: 18 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229619654

Digital Object Identifier
doi:10.7169/facm/1229619654

Mathematical Reviews number (MathSciNet)
MR2363827

Zentralblatt MATH identifier
1226.11105

Subjects
Primary: 11P05: Waring's problem and variants
Secondary: 11T55: Arithmetic theory of polynomial rings over finite fields 11P55: Applications of the Hardy-Littlewood method [See also 11D85]

Keywords
Waring's problem function fields

Citation

Liu, Yu-Ru; Wooley, Trevor D. The unrestricted variant of Waring's problem in function fields. Funct. Approx. Comment. Math. 37 (2007), no. 2, 285--291. doi:10.7169/facm/1229619654. https://projecteuclid.org/euclid.facm/1229619654


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