MULTIDIMENSIONAL RESIDUES AND POLYNOMIAL EQUATIONS

摘要

We describe some recent results of real algebraic geometry which, are obtained using multidimensional residues. Our main focus is on the algebraic formulas for topological invariants such as the mapping degree and Euler characteristic. Our proof of the algebraic formula for the mapping degree is based on the properties of a multidimensional logarithmic residue, which are discussed in detail. Several recent applications of the main results are also presented. In particular, we discuss applications to the topological study of quadratic mappings, configuration spaces, quadratic Poisson structures, and Gaussian random polynomials.