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, , 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 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, . 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 is a restricted sum of at most ten cubes. We also prove, in Theorem 9, that by adding to a given 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.
@article{bwmeta1.element.bwnjournal-article-aav92i2p109bwm,
author = {Luis Gallardo},
title = {On the restricted Waring problem over $\_{2^n}[t]$
},
journal = {Acta Arithmetica},
volume = {92},
year = {2000},
pages = {109-113},
zbl = {0948.11034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav92i2p109bwm}
}
Luis Gallardo. On the restricted Waring problem over $_{2^n}[t]$
. Acta Arithmetica, Tome 92 (2000) pp. 109-113. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav92i2p109bwm/
[00000] [1] M. Car et J. Cherly, Sommes de cubes dans l’anneau , Acta Arith. 65 (1993), 227-241. | Zbl 0789.11057
[00001] [2] R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1984, pp. 327 and 295.
[00002] [3] L. N. Vaserstein, Sums of cubes in polynomial rings, Math. Comp. 56 (1991), 349-357. | Zbl 0711.11013