Sums of fourth powers of polynomials over a finite field of characteristic 3
type: Item
instance of: scholarly article
title: Sums of fourth powers of polynomials over a finite field of characteristic 3
zbMATH Open document ID: 1213.11195
author: Mireille Car
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2009-10-26T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1229696539
review text: 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.
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 11P05
zbMATH DE Number: 5620907
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5620907