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September 2008 Sums of Fourth Powers of Polynomials over a~Finite Field of Characteristic 3
Mireille Car
Funct. Approx. Comment. Math. 38(2): 195-220 (September 2008). DOI: 10.7169/facm/1229696539

Abstract

Let $F$ be a finite field with $q$ elements and characteristic $3.$ A sum $$M = M_{1}^4+\ldots+ M_{s}^4$$ of fourth powers of polynomials $M_1,\dots, M_{s}$ is a strict one if $ 4\deg M_i < 4 + \deg M$ for each $i= 1,\ldots, s.$ Our main results are: Let $P\in F[T]$ of degree $\geq 329.$ If $q>81$ is congruent to $1$ (mod. $4$), then $P$ is the strict sum of $9$ fourth powers; if $q=81$ or if $q>3$ is congruent to $3$ (mod $4$), then $P$ is the strict sum of $10$ fourth powers. If $q=3,$ every $P\in F[T]$ which is a sum of fourth powers is a strict sum of $12$ fourth powers, if $q=9,$ every $P\in F[T]$ which is a sum of fourth powers and whose degree is not divisible by $4$ is a strict sum of $8$ fourth powers; every $P\in F[T]$ which is a sum of fourth powers, whose degree is divisible by $4$ and whose leading coefficient is a fourth power is a strict sum of $7$ fourth powers.

Citation

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Mireille Car. "Sums of Fourth Powers of Polynomials over a~Finite Field of Characteristic 3." Funct. Approx. Comment. Math. 38 (2) 195 - 220, September 2008. https://doi.org/10.7169/facm/1229696539

Information

Published: September 2008
First available in Project Euclid: 19 December 2008

zbMATH: 1213.11195
MathSciNet: MR2492856
Digital Object Identifier: 10.7169/facm/1229696539

Subjects:
Primary: 11T55
Secondary: 11P23

Keywords: finite fields , polynomials , Waring's Problem

Rights: Copyright © 2008 Adam Mickiewicz University

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Vol.38 • No. 2 • September 2008
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