Regularity at infinity of real mappings and a Morse-Sard theorem (Q2892843)

A real or complex polynomial map \(f: \mathbb K^n \to \mathbb K^p\) defines a locally trivial fibration over the complement of the set of atypicla values. The authors study the relations between several regularity conditions at infinity. The central object of the paper is \(t\)-regularity. It is shown that it is equivalent to the regularity condition of \textit{K. Kurdyka, P. Orro} and \textit{S. Simon} [J. Differ. Geom. 56, No.~1, 67--92 (2000; Zbl 1067.58031)], and also to the Malgrange condition of \textit{T. Gaffney} [Compos. Math. 119, No.~2, 157--167 (1999; Zbl 0945.32013)], here also formulated in the real case. Under a `fairness' condition there is also an interpretation in terms of integral closure. The authors show that \(t\)-regularity implies \(\rho_E\)-regularity (a Milnor-type condition of transversality to the Euclidean distance function), which in turn implies topological triviality at infinity.