The discriminant of a morphism of real algebraic manifolds (Q1378000)

Let \(W\) and \(V\) be real affine algebraic varieties and \(\Phi: W\to V\) a regular morphism. Assume \(V\) is irreducible. Then a rational function \(\delta_\Phi\) on \(V\) is called a discriminant of \(\Phi\) if there exists \(\varepsilon\in\mathbb{Z}/4\mathbb{Z}\) such that, for any \(x\in V\), the Euler-Poincaré characteristic \(\chi(\Phi^{-1}(x))\bmod 4\) is equal to \(\varepsilon\) if \(\delta_\Phi (x)>0\), and \(\chi(\Phi^{-1} (x))\bmod 4\) is equal to \(\varepsilon+2\) if \(\delta_\Phi (x)<0\). In this interisting paper it is shown the existence of a discriminant of \(V\). This result is closely related to the theory of algebraically constructible functions due to McCrory, Parusiński, Szafraniec.