Semi-Algebraic Functions with Non-Compact Critical Set

Abstract

AbstractLet $$X\subset {\mathbb {R}}^n$$ X ⊂ R n be a closed semi-algebraic set, $$F:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$$ F : R n → R be a $${\mathcal {C}}^2$$ C 2 semi-algebraic function and $$f=F_{|X}:X\rightarrow {\mathbb {R}}^n$$ f = F | X : X → R n be the restriction of F to X. We define the global index of a critical value $$c_i$$ c i of f and prove an index formula for $$\chi (X)$$ χ ( 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 $$\chi (X).$$ χ ( X ) .