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2000 | 92 | 2 | 109-113
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On the restricted Waring problem over $𝔽_{2^n}[t]$

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EN
Abstrakty
EN
1. Introduction. The Waring problem for polynomial cubes over a finite field F of characteristic 2 consists in finding the minimal integer m ≥ 0 such that every sum of cubes in F[t] is a sum of m cubes. It is known that for F distinct from 𝔽₂, 𝔽₄, $𝔽_{16}$, each polynomial in F[t] is a sum of three cubes of polynomials (see [3]).
If a polynomial P ∈ F[t] is a sum of n cubes of polynomials in F[t] such that each cube A³ appearing in the decomposition has degree < deg(P)+3, we say that P is a restricted sum of n cubes.
The restricted Waring problem for polynomial cubes consists in finding the minimal integer m ≥ 0 such that each sum of cubes S in F[t] is a restricted sum of m cubes.
The best known result for the above problem is that every polynomial in $𝔽_{2^n}[t]$ of sufficiently high degree that is a sum of cubes, is a restricted sum of eleven cubes. This result was obtained by the circle method in [1].
Here we improve this result using elementary methods. Let F be a finite field of characteristic 2, distinct from 𝔽₂, 𝔽₄, $𝔽_{16}$. In Theorem 7, we prove that every polynomial in F[t] is a restricted sum of at most nine cubes, and that every polynomial in $𝔽_{16}[t]$ is a restricted sum of at most ten cubes.
We also prove, in Theorem 9, that by adding to a given $P ∈ 𝔽_{2^n}[t]$ some square B² with deg(B²) < deg(P) + 2, the resulting polynomial is a restricted sum of at most four cubes, for all n ≠ 2.
Słowa kluczowe
Czasopismo
Rocznik
Tom
92
Numer
2
Strony
109-113
Opis fizyczny
Daty
wydano
2000
otrzymano
1998-02-20
poprawiono
1999-09-29
Twórcy
  • Department of Mathematics, University of Brest, 6, Avenue le Gorgeu, 29285, Brest Cedex, France
Bibliografia
  • [1] M. Car et J. Cherly, Sommes de cubes dans l'anneau $𝔽_{2^h}[X]$, Acta Arith. 65 (1993), 227-241.
  • [2] R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1984, pp. 327 and 295.
  • [3] L. N. Vaserstein, Sums of cubes in polynomial rings, Math. Comp. 56 (1991), 349-357.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-aav92i2p109bwm
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