Upper bounds of topology of complex polynomials in two variables

TL;DR

This paper establishes quantitative bounds on the topology of complex polynomials in two variables with specific conditions, providing tools for analyzing their critical values and related topological features.

Contribution

It introduces explicit upper bounds on coefficients, topology, and cycle lengths of such polynomials, advancing understanding of their critical value structure.

Findings

Upper bounds for sum of lower term coefficients

Bounds on size of bidisc containing topology

Limits on cycle lengths in homology groups

Abstract

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 $S_t=\{ H=t\}$; - the minimal lengths of representatives of cycles in $H_1(S_t,\zz)$ 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.