Mapping degree and Euler characteristic as signatures

We'll describe an effective method of computing various topological invariants in real algebraic context, based on the interpretation of mapping degree as the signature of a certain quadratic form of Gorenstein type. In particular, it will be shown that the Euler characteristic of an explicitly given compact algebraic variety can also be computed as the signature of an appropriate quadratic form, which gives a (sort of) multidimensional generalization of Sturm algorithm and leads to a variety of explicit formulae for the topological invariants of real algebraic varieties and isolated singularities developed by W.Bruce, T.Fukui, N.Dutertre Z.Szafraniec and the speaker. Moreover, the Euler characteristic of a compact semi-algebraic set can be calculated in a similar way, which yields an effective approach to several classical problems and a number of concrete new results. In particular, along these lines we'll describe the structure of zero-set and give an effective criterion of stability for unilateral quaternionic polynomials the need for which was explicitly stated in the recent physical literature on quaternionic quantum mechanics. As examples of other visual applications, we'll present exact estimates for the index of a polynomial vector field given by V.Arnold and A.Khovansky and an explicit formula for the expected value of index of a Gaussian random vector field of the type considered in papers of M.Shub and S.Smale on complexity of Bezout theorem. In conclusion, we'll outline the most recent application concerned with counting critical points of various energy functions on configuration spaces of mechanical linkages.