Semi-Algebraic Functions with Non-Compact Critical Set

Let X ⊂ R n be a closed semi-algebraic set, F : R n → R be a C 2 semi-algebraic function and f = F | X : X → 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 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 χ ( X ) .