Le problème de Waring pour les corps de fonctions. (Waring’s problem for function fields). (French) Zbl 0758.11040

Car, Mireille

Journées arithmétiques, Exp. Congr., Luminy/Fr. 1989, Astérisque 198-200, 77-82 (1991).

[For the entire collection see Zbl 0743.00058.]
The paper is concerned with the restricted Waring problem for rings \(O_ S\) of \(S\)-integers in algebraic function fields in one variable over finite fields \(F_ q\). Let \(k> 1\) be an integer. When \(O_ S = F_ q[X]\) one denotes by \(G(k)\) the smallest integer \(m\), if it exists, such that for every \( A\) in \(F_ q[X]\) of large enough degree \(N\) there are \(m\) polynomials \(A_ n\) of degree \( < N/k + 1\) such that \( A=A_ 1^ k + ... + A_ m^ k\). It has been known that \(G(k) \leq k2^{k-1}+ 1\) when \( k < p\), where \(p\) is the characteristic of \( F_ q\). The author extends the definition of \(G(k)\) from \( F_ q[X]\) to \(O_ S \) and proves the following generalization of the above estimate: \[ G(k) \leq 1+ k2^{k-1} \text{Card}(S) \qquad \text{when } k<p. \]

Reviewer: L.N.Vaserstein and E.R.Wheland (University Park)

MSC:

11P05	Waring’s problem and variants
11R58	Arithmetic theory of algebraic function fields
11T55	Arithmetic theory of polynomial rings over finite fields

Keywords:

sum of polynomials; restricted Waring problem; rings of \(S\)-integers; algebraic function fields

Citations:

Zbl 0743.00058