## Functiones et Approximatio Commentarii Mathematici

### Sums of Fourth Powers of Polynomials over a~Finite Field of Characteristic 3

Mireille Car

#### 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.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 38, Number 2 (2008), 195-220.

Dates
First available in Project Euclid: 19 December 2008

https://projecteuclid.org/euclid.facm/1229696539

Mathematical Reviews number (MathSciNet)
MR2492856

Zentralblatt MATH identifier
1213.11195

Subjects
Primary: 11T55: Arithmetic theory of polynomial rings over finite fields
Secondary: 11P23

#### Citation

Car, Mireille. Sums of Fourth Powers of Polynomials over a~Finite Field of Characteristic 3. Funct. Approx. Comment. Math. 38 (2008), no. 2, 195--220. https://projecteuclid.org/euclid.facm/1229696539

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