Critical point methods and effective real algebraic geometry: new results and trends

Critical point methods are at the core of the interplay between polynomial optimization and polynomial system solving over the reals. These methods are used in algorithms for solving various problems such as deciding the existence of real solutions of polynomial systems, performing one-block real quantifier elimination, computing the real dimension of the solution set, etc. The input consists of $s$ polynomials in $n$ variables of degree at most $D$. Usually, the complexity of the algorithms is $(s\, D)^{O(n^\alpha)}$ where $\alpha$ is a constant. In the past decade, tremendous efforts have been deployed to improve the exponents in the complexity bounds. This led to efficient implementations and new geometric procedures for solving polynomial systems over the reals that exploit properties of critical points. In this talk, we present an overview of these techniques and their impact on practical algorithms. Also, we show how we can tune them to exploit algebraic and geometric structures in two fundamental problems. The first one is real root finding of determinants of $n$-variate linear matrices of size $k\times k$. We introduce an algorithm whose complexity is polynomial in ${{n+k}\choose{k}}$ (joint work with S. Naldi and D. Henrion). This improves the previously known $k^{O(n)}$ bound. The second one is about computing the real dimension of a semi-algebraic set. We present a probabilistic algorithm with complexity $(s\, D)^{O(n)}$, that improves the long-standing $(s\, D)^{O(n^2)}$ bound obtained by Koi\-ran (joint work with E. Tsigaridas).