Semi-Algebraic Functions with Non-Compact Critical Set

Abstract

Let Xsubset {mathbb {R}}^n be a closed semi-algebraic set, F:{mathbb {R}}^nrightarrow {mathbb {R}} be a {mathcal {C}}^2 semi-algebraic function and f=F_{|X}:Xrightarrow {mathbb {R}}^n be the restriction of F to X. We define the global index of a critical value c_i of f and prove an index formula for chi (X) that generalizes a result previously proved by the authors for the case of isolated critical points. We define also new indices at infinity and prove an alternative index formula for chi (X).