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

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.