New Applications of Algebraic Formulae for Topological Invariants

Abstract

Some geometric results are presented which can be derived from algebraic formulae for topological degree and Euler characteristic. In particular, it is shown that Euler characteristics of configuration spaces and work spaces of mechanical linkages can be computed in an algorithmic way. We also find the expected gradient degree of rotation invariant Gaussian random polynomial on an even-dimensional space.

Complex Points of Two-Dimensional Surfaces

“…The main goal of this paper being the obtaining of an explicit formula for the number of complex points, we begin by recalling some results about the counting of real roots of polynomial equations. We basically follow [12] with slight modifications made to fit into the context of the present paper.…”

“…Finally, let us notice that under the assumption of finite multiplicity the origin is an isolated zero of f , in other words, 0 is isolated in f −1 (0). As it is well known, in this case one can define the local topological (mapping) degree deg 0 f [12]. Theorem 1.1 ([8], [11]).…”

“…As shown in [6], [12], the Euler characteristic of a compact algebraic set can be expressed through the local degree of an auxiliary polynomial mapping. We reproduce the corresponding general formula from [6].…”

“…As usual, the intersection number can be computed as the sum of local intersection indices. For a generic X, all complex points are transversal so in this case the total index is just the algebraic number of complex points [12].…”

“…Recently, it has turned out that some of the above topics can be successfully treated in the case where a surface is given by polynomial equations, using the so-called signature formulae for topological invariants [12]. For example, this approach is quite effective in the case of the graph of a polynomial planar endomorphism [13].…”