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Sums of cubes in the ring \(\mathbb{F}_{2^ h}[X]\). (Sommes de cubes dans l’anneau \(\mathbb{F}_{2^ h}[X]\).) (French) Zbl 0789.11057

Let \(q\) be a power of a prime number \(p\). Let \(k \geq 2\) be an integer. If \(M\), \(M_ 1,\dots,M_ s\) are polynomials in the ring \(\mathbb{F}_ q[X]\) such that (i) \(M=M_ 1^ k + \cdots+M_ s^ k\), (ii) \(\deg (M_ i) \leq n\) if \(k(n-1)< \deg M \leq kn\), we say that (i) is a strict representation of \(M\) as a sum of \(s\) \(k\)th powers. The existence of such representations was known for exponents \(k<p\). Here, we prove the existence of such representations in the case \(k=3\), \(p=2\). More precisely, we prove the following result:
Let \({\mathcal M}_ q\) be the subset of \(\mathbb{F}_ q [X]\) defined as follows. \({\mathcal M}_ q=\mathbb{F}_ q [X]\) if \(q \geq 8\), \({\mathcal M}_ 2\) is the set of polynomials \(M \in \mathbb{F}_ 2 [X]\) congruent to 0 or 1 mod \(X^ 2+X+1\), \({\mathcal M}_ 4\) is the set of polynomials \(M \in \mathbb{F}_ 4 [X]\) congruent to 0 or 1 mod polynomials of degree 1, and such that, either 3 does not divide \(\deg M\), or 3 divides \(\deg M\) and \(M\) is monic. Let \(G(q,3)\) be the least integer \(s\), if it exists, such that every polynomial \(M \in {\mathcal M}_ q\) whose degree is large enough, admits a strict representation as a sum of \(s\) cubes. Then, \[ G(q,3) \leq 11. \]
Reviewer: M.Car (Marseille)

MSC:

11P05 Waring’s problem and variants
11T06 Polynomials over finite fields
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