Une estimation asymptotique du nombre de solutions approchées d'une équation p-adique. (Asymptotic estimation of the number of approximate solutions of a p-adic equation)
Let p be a prime number, \({\mathbb{Q}}_ p\) the field of p-adic numbers, K a finite extension of \({\mathbb{Q}}_ p\), R the ring of integers of K, P its maximal ideal and \(q=card(R/P)\), and \(f\in R[X_ 1,...,X_ n]\) a polynomial such that \(f(0)=0\). Let \(N_ i\) be the number of solutions of the equation \(f(x)=0\) in \(R/P^ iR.\) The author proves that if f has an isolated singularity non-rational in the origin, then \[ \limsup (N_ i)^{1/i} \leq q^{(n-\sum^{n- 1}_{i=0}(1+\theta^ i(f))^{-1})}, \] where \(\theta^ i(f)\) are the polar invariants defined by B. Teissier. This result generalizes the curve case obtained by \textit{J. Igusa} [Complex Anal. algebr. Geom., Collect. Pap. dedic. K. Kodaira, 357-368 (1977; Zbl 0355.14012)]. Also some estimates are given in terms of the Newton polyhedra of f. The author uses Hodge theory.
scientific article; zbMATH DE number 3981291
scientific article