# 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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topLuis 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 -

## References

top- [1] M. Car et J. Cherly, Sommes de cubes dans l’anneau ${}_{{2}^{h}}\left[X\right]$, Acta Arith. 65 (1993), 227-241. Zbl0789.11057
- [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. Zbl0711.11013

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