On the fiber of a polynomial with isolated critical points (Q1965133)

Let~\(f: \mathbb{R}^n\rightarrow \mathbb{R}\) be a polynomial map such that its first-order partial derivatives generate a zero-dimensional ideal. In an earlier paper [\textit{N. Dutertre}, J. Pure Appl. Algebra 139, No. 1-3, 41-60 (1999)], the author established, among other things, a formula for the Euler characteristic of the fiber~\(f^{-1}(0)\) in terms of the signature of a quadratic form, if~\(f\) is a proper map (loc. cit., theorem~5.2). In the present paper, this formula is generalized to not necessarily proper polynomial maps. The main ingredients of the proof are the global residue [see \textit{E. Becker, J. P. Cardinal, M.-F. Roy} and \textit{Z. Szafraniec} in: Algorithms in algebraic geometry and applications. Proc. MEGA-94 Conf., Santander 1994, Prog. Math. 143, 79-104 (1996; Zbl 0873.13013)] and Morse theory for manifolds with boundary [\textit{H. A. Hamm} and \textit{Lê Dũng Tráng}, J. Reine Angew. Math. 389, 157-189 (1988; Zbl 0646.14012)].