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On the restricted Waring problem over $𝔽_{2^n}[t]$

100%

Gallardo L.

Acta Arithmetica

2000

tom 92

nr 2

109-113

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.

Sums of cubes of polynomials

63%

Car M., Gallardo L.

Acta Arithmetica

2004

tom 112

nr 1

41-50

Ograniczanie wyników

2 Acta Arithmetica

2 Gallardo L.

1 Car M.

1 2004

1 2000

Wyniki wyszukiwania

On the restricted Waring problem over $𝔽_{2^n}[t]$

Sums of cubes of polynomials