Sums of fourth powers of polynomials over a finite field of characteristic 3. (English) Zbl 1213.11195

Car, Mireille

Funct. Approximatio, Comment. Math. 38, Part 2, 195-220 (2008).

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

Reviewer: L. N. Vaserstein (University Park)

MSC:

11T55	Arithmetic theory of polynomial rings over finite fields
11P05	Waring’s problem and variants

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