2005/09/30 by Alexey Glutsyuk, Glutsyuk, Alexey
Mathematics · #14D05 #30C10 #30C15 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #math.AG #math.CV #msc:14D05 #msc:30C10 #msc:30C15
paper · pdf · doi:10.48550/arxiv.math/0509727
51 pages
arxiv created 2005/09/30 · openalex publication_date 2005/09/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper deals with a complex polynomial H in two variables having - a generic highest homogeneous part (without multiple zero lines), - nonconstant lower terms. In particular, under these conditions the polynomial H has at least two distinct critical values. We prove quantitative versions of this statement. Supposing H appropriately normalized (by affine coordinate changes in the image and in the source) we prove upper bounds for the following quantities: - the sum of the coefficients of the lower terms; - the minimal size of a bidisc containing all the nontrivial topology of a given level curve St=\ H=t\; - the minimal lengths of representatives of cycles in H1(St,\zz) vanishing along appropriate paths from t to the critical values of H; - the intersection indices of the latter cycles. All these results (expect for the latter bound) are used in my joint work with Yu.S.Ilyashenko "Restricted version of the Hilbert 16-th problem" (available on the arxiv). In the latter paper we obtain an explicit upper bound of the number of zeros for a wide class of Abelian integrals.