# On the restricted Waring problem over ${}_{{2}^{n}}\left[t\right]$

Acta Arithmetica (2000)

• Volume: 92, Issue: 2, page 109-113
• ISSN: 0065-1036

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## Abstract

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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 ₂, ₄, ${}_{16}$, 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 ${}_{{2}^{n}}\left[t\right]$ 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 ₂, ₄, ${}_{16}$. 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 ${}_{16}\left[t\right]$ is a restricted sum of at most ten cubes. We also prove, in Theorem 9, that by adding to a given $P{\in }_{{2}^{n}}\left[t\right]$ 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.

## How to cite

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Luis Gallardo. "On the restricted Waring problem over $_{2^n}[t]$." Acta Arithmetica 92.2 (2000): 109-113. <http://eudml.org/doc/207373>.

@article{LuisGallardo2000,
abstract = {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 ₂, ₄, $_\{16\}$, 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 $_\{2^n\}[t]$ 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 ₂, ₄, $_\{16\}$. 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 $_\{16\}[t]$ is a restricted sum of at most ten cubes. We also prove, in Theorem 9, that by adding to a given $P ∈ _\{2^n\}[t]$ 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.},
author = {Luis Gallardo},
journal = {Acta Arithmetica},
keywords = {Polynomials; Waring problem},
language = {eng},
number = {2},
pages = {109-113},
title = {On the restricted Waring problem over $_\{2^n\}[t]$},
url = {http://eudml.org/doc/207373},
volume = {92},
year = {2000},
}

TY - JOUR
AU - Luis Gallardo
TI - On the restricted Waring problem over $_{2^n}[t]$
JO - Acta Arithmetica
PY - 2000
VL - 92
IS - 2
SP - 109
EP - 113
AB - 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 ₂, ₄, $_{16}$, 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 $_{2^n}[t]$ 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 ₂, ₄, $_{16}$. 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 $_{16}[t]$ is a restricted sum of at most ten cubes. We also prove, in Theorem 9, that by adding to a given $P ∈ _{2^n}[t]$ 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.
LA - eng
KW - Polynomials; Waring problem
UR - http://eudml.org/doc/207373
ER -

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