[Paper Review] On the topology of rational functions in two complex variables

Thang Nguyen|arXiv (Cornell University)|Apr 24, 2012

Advanced Differential Equations and Dynamical Systems2 references3 citations

TL;DR

This paper characterizes critical values at infinity for rational functions in two complex variables using the Euler characteristic, Malgrange condition, and M-tameness. It proves that for rational functions with deg(f) > deg(g), the bifurcation set includes critical values at infinity, critical values of the numerator-denominator, and an additional set K₁(F), with equivalence of Malgrange and M-tameness conditions for characterizing these values.

ABSTRACT

We give some characterizations for the critical values at infinity of a rational function in two complex variables in terms of the Euler characteristic, the Malgrange condition and the M-tameness

Motivation & Objective

• To characterize the set of critical values at infinity, $ B_{ u}(F) $, for rational functions $ F = f/g $ in two complex variables.
• To extend known characterizations of critical values at infinity—previously valid only for polynomial functions—to the broader class of rational functions.
• To investigate the relationship between the Malgrange condition, M-tameness, and the Euler characteristic in determining $ B_{ u}(F) $.
• To demonstrate that the Fedoryuk condition is insufficient to characterize $ B_{ u}(F) $, providing counterexamples where $ \widetilde{K}_{ u}(F) \not\subset B_{ u}(F) $.

Proposed method

• Define $ K_1(F) $ as the set of values $ t_0 $ where the Milnor number of $ f - t_0 g $ at points in the base locus $ A(F) $ changes under small perturbations.
• Use the global Milnor fibration and the topology of fibers $ F^{-1}(t) $ to analyze the behavior at infinity.
• Apply the Curve Selection Lemma to sequences $ (x_k, y_k) \to \infty $ with $ F(x_k) \to t_0 $ and $ \|\mathrm{grad}F(x_k)\| \to 0 $, linking this to the Malgrange condition.
• Prove that if $ F $ satisfies the Malgrange condition at $ t_0 $, then it is M-tame at $ t_0 $, establishing a hierarchy between these conditions.
• Construct a smooth vector field on $ F^{-1}(D_\delta(t_0)) \setminus B_R $ with $ \langle v, x \rangle = 0 $ and $ \langle v, \mathrm{grad}F \rangle = 1 $, ensuring trivialization of the fibration.
• Use the projective compactification of fibers via $ G(x,y,z,t) = z^{d_t} f(x/z, y/z) - t z^{d_t} g(x/z, y/z) $ to analyze singularities at infinity.

Experimental results

Research questions

• RQ1 How can the critical values at infinity $ B_{\infty}(F) $ of a rational function $ F = f/g $ in two complex variables be characterized beyond the classical polynomial case?
• RQ2 What is the role of the Euler characteristic of the fiber $ F^{-1}(t) $ in determining whether $ t_0 \in B_{\infty}(F) $ for rational functions?
• RQ3 To what extent do the Malgrange condition and M-tameness coincide for rational functions, and how do they relate to $ B_{\infty}(F) $?
• RQ4 Can the Fedoryuk condition $ t \notin \widetilde{K}_{\infty}(F) $ be used to characterize $ B_{\infty}(F) $, or are there counterexamples?
• RQ5 What is the significance of the set $ K_1(F) $, and how does it contribute to the full bifurcation set $ B(F) $ when $ \deg f > \deg g $?

Key findings

• For rational functions $ F = f/g $ with $ \deg f > \deg g $, the full bifurcation set is $ B(F) = B_{\infty}(F) \cup K_0(F) \cup K_1(F) $, where $ K_1(F) $ captures changes in Milnor numbers at base points.
• The Euler characteristic of the fiber $ F^{-1}(t) $ is not constant in any neighborhood of $ t_0 $ if and only if $ t_0 \in B_{\infty}(F) $, generalizing a result from polynomial functions.
• The Malgrange condition implies M-tameness: if $ t_0 \notin K_{\infty}(F) $, then $ t_0 \notin M_{\infty}(F) $, establishing a hierarchy between these conditions.
• For $ t_0 \notin K_0(F) \cup K_1(F) $, the three conditions $ t_0 \in B_{\infty}(F) $, $ t_0 \in K_{\infty}(F) $, and $ t_0 \in M_{\infty}(F) $ are equivalent.
• Counterexamples are constructed where $ t_0 \in \widetilde{K}_{\infty}(F) $ but $ t_0 \notin B_{\infty}(F) $, showing that the Fedoryuk condition is not sufficient to characterize $ B_{\infty}(F) $.
• The result extends to the case $ \deg f < \deg g $ when $ t_0 \neq 0 $, with the same equivalence of conditions holding for $ B_{\infty}(F) $.