Multidimensional generalization of the Il'yashenko theorem on abelian integrals
type: Item
instance of: scholarly article
title: Multidimensional generalization of the Il'yashenko theorem on abelian integrals
zbMATH Open document ID: 0930.37036
DOI: 10.1007/BF02466015
publication date: 1998-10-11T00:00:00Z
review text: The main theorem proved in this paper is as follows: Let \(P(x_{1},\dots,x_k\) be a real polynomial of degree \(n>k\) with all complex critical points being Morse and all critical values distinct. Let \(P=0\) be a noncritical level surface and \(\alpha\) a polynomial \((k-1)\)-form with coefficients of degree \(\leq n-k+1\). Let \(\gamma(0)\) be a compact connected component of the real surface \(P=0\) and \(\gamma(c) \subset (P=0)\) its continuous deformation. If the identity \(\int_{\gamma(c)} \alpha \equiv 0\) holds for all sufficiently small values of parameter \(c\), then \(\alpha\) is exact. This result generalizes a theorem of Il'yashenko on the zeros of Abelian integrals to higher dimensions. The approach adopted by the author is based on singularity theory and algebraic geometry which differs from Il'yashenko's.
reviewed by: Na Jisheng
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 14K20
Mathematics Subject Classification ID: 37E35
Mathematics Subject Classification ID: 32S70
zbMATH DE Number: 1179578
zbMATH Keywords: abelian integral
zbMATH Keywords: polynomial form
zbMATH Keywords: exact form
zbMATH Keywords: singularity
author: I. A. Khovanskaya
MaRDI profile type: Publication
cites work: Deformations of holomorphic foliations having a meromorphic first integral.
cites work: A TORELLI THEOREM FOR ALGEBRAIC SURFACES OF TYPEK3
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1179578