On the generalized critical values of a polynomial mapping
Let \(f: k^n \to k^m\) be a polynomial dominant mapping, where \(k= {\mathbb R}\) or \(k= {\mathbb C}\). It is known that \(f\) is a locally trivial fibration outside the bifurcation set \(B(f)\), which has measure 0. The set \(K(f)\) of generalized critical values of \(f\) is defined as \(K(f) = K_0(f) \cup K_\infty(f)\), where \(K_0(f)\) is the set of classical critical values of \(f\) and \(K_\infty(f)\) is the set of asymptotic critical values at infinity. This set \(K(f)\) is known to be a semialgebraic set (or algebraic in the complex case) [ see the paper by \textit{K. Kurdyka, P. Orro} and \textit{S. Simmon}, J. Differ. Geom. 56, 67--92 (2000; Zbl 1067.58031)]. One of the results of the paper under review is that \(B(f) \subset K(f)\), when \(f\) is a \(k\)-smooth mapping. This is a particular case of a more general result of \textit{P. J. R. Rabier} [ Ann. Math. 146, 647--691 (1997; Zbl 0919.58003)]. The interest in the proof given here is that it is direct and short. The main result of the paper states that if \(f=(f_1, \ldots,f_m): {\mathbb C}^n \to {\mathbb C}^m\) is a polynomial dominant mapping then the set \(K(f)\) is a proper algebraic subset of \({\mathbb C}^m\) and it is contained in a hypersurface of degree at most \(D = (d+s(m-1)(d-1))^n\), where \(s = {n\choose m} \), and \(\text{ deg} \, f_i \leq d\). An algorithm to compute the set \(K(f)\) effectively is given in the last section by using Gröbner basis.
scientific article; zbMATH DE number 1922375