[Paper Review] Bifurcation values and monodromy of mixed polynomials

Ying Chen, Mihai Tibăr|arXiv (Cornell University)|Nov 22, 2010

Advanced Differential Equations and Dynamical Systems19 references13 citations

TL;DR

This paper establishes a real counterpart to complex polynomial bifurcation theory by introducing ρ-regularity at infinity for mixed polynomials, proving that the bifurcation set is bounded and estimating it via Newton non-degenerate faces. The key result is a monodromy stability theorem for families of Newton strongly non-degenerate mixed polynomials with fixed Newton boundary at infinity, extending holomorphic results to the real setting despite non-dense and non-connected parameter spaces.

ABSTRACT

We study the bifurcation values of real polynomial maps $f: \bR^{2n} o \bR^2$ which reflect the lack of asymptotic regularity at infinity. We formulate real counterparts of some structure results which have been previously proved in case of complex polynomials by Kushnirenko, N\'emethi and Zaharia and other authors, emphasizing the typical real phenomena that occur.

Motivation & Objective

• To extend complex polynomial bifurcation theory to real mixed polynomials, where asymptotic regularity at infinity is not guaranteed.
• To characterize the bifurcation locus B(f) for real polynomial maps f: R^{2n} → R^2 using ρ-regularity at infinity.
• To provide effective estimates of the bifurcation set S(f) using Newton non-degeneracy and bad faces of the Newton polyhedron.
• To prove monodromy stability at infinity for families of mixed polynomials with constant Newton boundary at infinity, despite the non-dense and non-connected nature of the Newton strongly non-degenerate condition in the real setting.

Proposed method

• Introduces ρ-regularity at infinity as a real analogue of Milnor's local condition, defining the Milnor set M(f) and the set S(f) of asymptotic bad values.
• Uses the semi-algebraic Sard-type theorem to show that S(f) ⊂ f(Sing f) ∪ {0} ∪ ∪_{Δ∈B} f_Δ(Sing f_Δ ∩ C*^n) for Newton non-degenerate mixed polynomials.
• Applies the curve selection lemma and face analysis to prove boundedness of f(Sing f) and S(f) under Newton strong non-degeneracy.
• Establishes a global fibration at infinity via transversality of fibres to large spheres, using the boundedness of critical and bad value sets.
• Applies Ehresmann's fibration theorem to show isotopy of monodromy fibrations across a 1-parameter family of mixed polynomials with fixed Newton boundary at infinity.
• Proves monodromy stability by constructing a continuous family of mixed polynomials with constant Newton boundary and showing the monodromy fibrations are isotopic across the family.

Experimental results

Research questions

• RQ1 How can the bifurcation locus B(f) of a real mixed polynomial f: R^{2n} → R^2 be estimated when asymptotic regularity fails at infinity?
• RQ2 What is the role of Newton non-degeneracy and bad faces in characterizing the set S(f) of asymptotic bad values in the real mixed setting?
• RQ3 Can monodromy at infinity be defined and stabilized in the real mixed polynomial setting, despite the lack of density and connectedness of the Newton strongly non-degenerate condition?
• RQ4 How does the monodromy at infinity behave under continuous deformation of mixed polynomials with fixed Newton boundary at infinity?
• RQ5 What new phenomena arise in the real mixed setting that do not occur in the holomorphic case, such as the contribution of non-degenerate faces to the bifurcation locus?

Key findings

• The bifurcation set B(f) is contained in S(f) ∪ f(Sing f), where S(f) is a semi-algebraic set of asymptotic bad values, and S(f) is bounded under Newton strong non-degeneracy.
• For Newton non-degenerate mixed polynomials, S(f) ⊂ {0} ∪ ∪_{Δ∈B} f_Δ(Sing f_Δ ∩ C*^n), providing an effective estimate of the bifurcation locus.
• The monodromy at infinity exists and is well-defined for Newton strongly non-degenerate mixed polynomials, with the image of f containing the complement of a disk in R^2.
• Monodromy at infinity is stable under continuous deformation of mixed polynomials with fixed Newton boundary at infinity, as shown via isotopy of fibrations.
• The monodromy of two Newton strongly non-degenerate mixed polynomials with the same Newton boundary at infinity and holomorphic (or anti-holomorphic) restrictions at infinity are isotopic.
• A new phenomenon is observed: a face can be non-degenerate in the Newton boundary but still contribute to the bifurcation locus, which cannot occur in the holomorphic setting.