Section: New Results
Critical Point Computations on Smooth Varieties: Degree and Complexity Bounds
Participant : Pierre-Jean Spaenlehauer [contact] .
This is a joint work with Mohab Safey El Din (Univ. Paris 6, EPI Polsys). This work led to a publication in the proceedings of the ISSAC conference [13].
Let be an equidimensional algebraic set and be an -variate polynomial with rational coefficients. Computing the critical points of the map that evaluates at the points of is a cornerstone of several algorithms in real algebraic geometry and optimization. Under the assumption that the critical locus is finite and that the projective closure of is smooth, we provide sharp upper bounds on the degree of the critical locus which depend only on and the degrees of the generic polar varieties associated to . Using these degree bounds and an algorithm due to Bank, Giusti, Heintz, Lecerf, Matera and Solernó, we derive complexity bounds which are quadratic in the degree bounds (up to logarithmic factors) and polynomial in all the other parameters of the problem.