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Let V $\subset$ C n be an equidimensional algebraic set and g be an n-variatepolynomial with rational coefficients. Computing the critical points of the mapthat evaluates g at the points of V is a cornerstone of several algorithms inreal algebraic geometry and optimization. Under the assumption that thecritical locus is finite and that the projective closure of V is smooth, weprovide sharp upper bounds on the degree of the critical locus which dependonly on deg(g) and the degrees of the generic polar varieties associated to V.Hence, in some special cases where the degrees of the generic polar varietiesdo not reach the worst-case bounds, this implies that the number of criticalpoints of the evaluation map of g is less than the currently known degreebounds. We show that, given a lifting fiber of V , a slight variant of analgorithm due to Bank, Giusti, Heintz, Lecerf, Matera and Solern{\'o} computesthese critical points in time which is quadratic in this bound up tologarithmic factors, linear in the complexity of evaluating the input systemand polynomial in the number of variables and the maximum degree of the inputpolynomials.

Critical Point Computations on Smooth Varieties: Degree and Complexity bounds