2001/12/15 by Alexey Glutsyuk, A. A. Glutsyuk, Glutsyuk, A. A. +3
Mathematics · #1 14K20 #34C05 #58F2 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.CV #math.DS #msc:1 #msc:14K20 #msc:34C05 #msc:58F2
paper · pdf · doi:10.48550/arxiv.math/0112156
45 pages
openalex publication_date 2001/12/15 · arxiv created 2005/09/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper deals with the \it infinitesimal Hilbert 16th problem: to find an upper estimate of the number of zeros of an Abelian integral regarded as a function of a parameter. In more details, consider a real polynomial H of degree n+1 in the plane, and a continuous family of ovals γt (compact components of level curves H = t) of this polynomial. Consider a polynomial 1-form ω with coefficients of degree at most n. Let I(t) = ∫γt ω. The problem is to give an upper estimate of the number of zeros of this integral. We solve a \it restricted version of this problem. Namely, the form ω is \it arbitrary,, and the polynomial H, though having an arbitrary degree, is not too close to the hypersurface of degenerate (non ultra-Morse) polynomials. We hope that the solution of the restricted version of the problem is a step to the solution of the complete (nonrestricted) version.