[Paper Review] Upper bounds of topology of complex polynomials in two variables

Alexey Glutsyuk|ArXiv.org|Sep 30, 2005

Advanced Differential Equations and Dynamical Systems2 references3 citations

TL;DR

This paper establishes explicit upper bounds for the topology of complex algebraic curves defined by polynomials in two variables, focusing on critical values, vanishing cycles, and the period determinant of Abelian integrals. Using normalization, Bezout's theorem, and norm estimates, it derives quantitative bounds on cycle lengths, intersection indices, and integrals, culminating in a lower bound for the period determinant essential for studying zeros of Abelian integrals in dynamical systems and singularity theory.

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.

Motivation & Objective

• To provide explicit upper bounds for the topological complexity of level curves of complex polynomials in two variables.
• To establish quantitative estimates for the lengths of canonical representatives of vanishing cycles and their intersection indices.
• To derive upper bounds for the absolute values of Abelian integral matrix entries and a lower bound for the period determinant.
• To support applications in the study of zeros of Abelian integrals over real ovals of polynomial level curves.
• To generalize results for ultra-Morse polynomials without assuming reality of the polynomial or its forms.

Proposed method

• Normalize the polynomial via affine transformations to ensure the highest-degree homogeneous part is generic and the origin is a critical point.
• Use Bezout's theorem and a priori bounds on the degree and coefficients to estimate the topology of unit-scaled polynomials.
• Apply max-norm and Hermitian norm estimates to control the size of coefficients and the norm of linear operators associated with the polynomial.
• Derive lower bounds for the maximal critical value of the projection map to control the location of critical values.
• Use integral estimates and asymptotic inequalities (e.g., Stirling-type bounds on logarithmic sums) to estimate the constant $ C_n $ in the period determinant formula.
• Combine bounds on discriminants, coefficient norms, and the constant $ C(h, ar{ heta}) $ to derive the final lower bound for the period determinant $ \Delta(t) $.

Experimental results

Research questions

• RQ1 What are the explicit upper bounds for the topology of level curves of complex polynomials in two variables under appropriate normalization?
• RQ2 How can one bound the lengths of canonical representatives of vanishing cycles in the homology of level curves?
• RQ3 What are the quantitative estimates for the intersection indices of vanishing cycles in the homology of the level curves?
• RQ4 How can one bound the absolute values of the entries of the Abelian integral matrix associated with monomial 1-forms?
• RQ5 What is the sharp lower bound for the period determinant $ \Delta(t) = \det \mathbb{I}(t) $, and how does it depend on the homogeneous part and the forms?

Key findings

• The paper establishes an upper bound for the topology of level curves of complex polynomials, showing that the number of critical points and topological complexity are controlled by the degree and normalization.
• It proves that the lengths of canonical representatives of vanishing cycles are bounded by $ O(n^2) $ in terms of the degree $ n $, with explicit dependence on the geometry of the critical values.
• The intersection indices of vanishing cycles are bounded by $ O(n^4) $, derived from the number of pieces in their projections and topological constraints.
• The absolute values of the entries of the Abelian integral matrix $ \mathbb{I}(t) $ are bounded above by $ \exp(O(n^2)) $, using bounds on cycle lengths and coefficient norms.
• A sharp lower bound for the period determinant is established: $ |\Delta(t)| > n^{-60n^3} (c')^{6n^3} $, where $ c' $ is a constant depending on the homogeneous part.
• The constant $ C_n $ in the period determinant formula satisfies $ \ln |C_n| > -12n^2 $, which is essential for the lower bound of the determinant and supports applications in bifurcation theory and Abelian integrals.