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)
type: Item
instance of: scholarly article
title: 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)
zbMATH Open document ID: 0607.12008
DOI: 10.1007/BF01388790
author: François Loeser
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143357
review text: 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.
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 12E12
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 3981291
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: number of roots of p-adic equation
zbMATH Keywords: polar invariants
zbMATH Keywords: Newton polyhedra
zbMATH Keywords: Hodge theory
reviewed by: Marcel Morales
MaRDI profile type: Publication
cites work: Q3718837
cites work: Stable structures on manifolds. I: Homeomorphismus of \(S^ n\). II: Stable manifolds. III: Applications
cites work: Complex powers and asymptotic expansions. I. Functions of certain types.
cites work: Some Observations on Higher Degree Characters
cites work: Lectures on forms of higher degree. Notes by S. Raghavan
cites work: Q3719034
cites work: Some local consequences of Hodge theory
cites work: Semicontinuity of the singularity spectrum
cites work: Variétés polaires. I: Invariants polaires des singularites d'hypersurfaces
cites work: Newton polyhedra and estimation of oscillating integrals
cites work: Q3933097
cites work: The complex exponent of a singularity does not change along strata mu=const
OpenAlex ID: W2091619878
Recommended article: Q5835054
Recommended article: Q5848299
Recommended article: An estimate of approximation constants for \(p\)-adic and real varieties
Recommended article: Q3695331
Recommended article: On approximation of \(p\)-adic numbers by \(p\)-adic algebraic numbers
Recommended article: Q4090417
Recommended article: Q3331285
Recommended article: Q5835053
Recommended article: Q5599271
Recommended article: On diophantine approximations in a field ofp-adic numbers
description: scientific article; zbMATH DE number 3981291