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 $V\subset {ℂ}^n$ be an equidimensional algebraic set and $g$ be an $n$-variate polynomial with rational coefficients. Computing the critical points of the map that evaluates $g$ at the points of $V$ 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 $V$ is smooth, we provide sharp upper bounds on the degree of the critical locus which depend only on $deg\left(g\right)$ and the degrees of the generic polar varieties associated to $V$. 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.