Remarks on Quadratic Mappings

Abstract

We present several results concerning the geometry and topology of quadratic mappings. The main attention is given to certain basic properties, namely, properness, surjectivity, stability, and topology of fibers. The structure of singular sets, discriminants, bifurcation diagrams, and Pareto sets is also discussed. After describing the setting and algebraic methods for computing the topological degree and Euler characteristic that are crucial for our approach, we concentrate on the study of quadratic endomorphisms and quadratic mappings into the plane. We begin by considering homogeneous quadratic endomorphisms of the plane and give criteria of properness and possible values of topological degree, as well as some geometric information about the structure of singular sets and discriminants. Next, we deal with analogs of the above results for proper quadratic endomorphisms in arbitrary dimension. In particular, we obtain an explicit estimate for the topological degree of quadratic endomorphism in terms of dimension and present examples showing that this estimate is exact. After this we discuss homogeneous quadratic mappings from ℝn into the plane and obtain a number of results on the Euler characteristic and topology of fibers. Finally, we derive some corollaries in the case of numerical range mapping of a complex square matrix.