Let X ⊂ Cn be an equidimensional complex algebraic set and let f : X → C be a polynomial function. For each c ∈ C, we define the global Brasselet number of f at c, a global counterpart of the Brasselet number defined by the authors in a previous work, and the Brasselet number at infinity of f at c. Then we establish several formulas relating these numbers to the topology of X and the critical points of f.