We give a characterization of the irreducible components of a Weierstrass-type (W-type) analytic (resp. algebraic, Nash) variety in terms of the orbits of a Galois group associated in a natural way to this variety. Since every irreducible variety of pure dimension is (locally) a component of a W-type variety, this description may be applied to any such variety.
We prove that the semialgebraic, algebraic, and algebraic nonsingular points of a definable set in o-minimal structure with analytic cell decomposition are definable. Moreover, the operation of taking semialgebraic points is idempotent and the degree of complexity of semialgebraic points is bounded.
For a symmetric (= invariant under the action of a compact Lie group G) semialgebraic basic set C, described by s polynomial inequalities, we show, that C can also be written by s + 1 G-invariant polynomials. We also describe orbit spaces for the action of G by a number of inequalities only depending on the structure of G.
Let Y be a real algebraic subset of and be a polynomial map. We show that there exist real polynomial functions on such that the Euler characteristic of fibres of is the sum of signs of .
The paper is concerned with an effective formula for the Euler characteristic of the link of a weighted homogeneous mapping with an isolated singularity. The formula is based on Szafraniec’s method for calculating the Euler characteristic of a real algebraic manifold (as the signature of an appropriate bilinear form). It is shown by examples that in the case of a weighted homogeneous mapping it is possible to make the computer calculations of the Euler characteristics much more effective.
The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field ℝ of reals. A () smooth definable function φ: U → ℝ on an open set U in ℝⁿ determines two closed subsets W := u ∈ U: φ(u) ≤ 0, Z := u ∈ U: φ(u) = 0. We shall investigate the links of the sets W and Z at the points u ∈ U, which are well defined up to a definable homeomorphism. It is proven...
The paper is concerned with the relations between real and complex topological invariants of germs of real-analytic functions. We give a formula for the Euler characteristic of the real Milnor fibres of a real-analytic germ in terms of the Milnor numbers of appropriate functions.