Journal de Th´
eorie des Nombres
de Bordeaux
19
(2007), 205–219
On
p
-adic zeros of systems of diagonal forms
restricted by a congruence condition
par
Hemar GODINHO
et
Paulo H. A. RODRIGUES
R´esum´e. Cet article ´etudie l’existence de solutions non triviales
en entiersp-adiques de syst`emes d’´equations pour des formes ad-ditives. En supposant que l’´equationaxk
+byk
+czk
≡d(modp) ait une solution telle quexyz 6≡0 (modp), nous montrons qu’un syst`eme quelconque de formes additives de degr´ek et d’au moins 2·3R−1
·k+ 1 variables poss`ede toujours des solutionsp-adiques non-triviales, sip ∤ k. L’hypoth`ese ci-dessus pour l’existence de solutions non-triviales de l’´equation est v´erifi´ee si, par exemple, p > k4.
Abstract. This paper is concerned with non-trivial solvability
in p-adic integers of systems of additive forms. Assuming that the congruence equation axk+byk+czk ≡ d(modp) has a so-lution with xyz 6≡0 (modp) we have proved that any system of R additive forms of degree k with at least 2·3R
−1·k+ 1 vari-ables, has always non-trivialp-adic solutions, provided p∤k. The assumption of the solubility of the above congruence equation is guaranteed, for example, ifp > k4.
HemarGodinho
Departamento de Matem´atica Universidade de Bras´ılia 70.910-900, Bras´ılia, DF, Brasil E-mail:[email protected]
Paulo H. A.Rodrigues
Instituto de Matem´atica e Estat´ıstica Universidade Federal de Goi´as 74.001-970, Goiˆania, GO, Brasil E-mail:[email protected]
Manuscrit re¸cu le 21 octobre 2005.
