p-adic zeros of polynomials.

Hovanskii [HA] (generalizing a theorem of Descartes) has proved that there is a bound b(n, N) depending only on n and N (and not on the degrees of the F£) such that if such a System has only finitely many Solutions in US then it has at most b(n,N) Solutions in ffS. A p-adic analogue of this has been given in [DD], that for each p there is a bound ßp(n, N) such that if (*) has only finitely many Solutions (Jc l 9 . . . , x„) in the padic integers with the xt = l modp then it has at most ßp(n, N) such Solutions. In [DD] it is asked whether this bound can be taken independent of p. The main result of this paper is an affirmative ans wer to this question (Theorem 1). Let Cp be the completion of the algebraic closure of Qp (the p-adic numbers) and let Rp be the ring of integers in Cp. We also prove the analogous result for the rings Rp (Theorem 2). Let Zp be the set of units in Zp.