“Computing the dimension of a real algebraic set”, accepted at ISSAC 2021.
I will give a 6-lecture course entitled Computing with integrals in nonlinear algebra. Virtually held at MPI Leipzig, it will be given on zoom.
Schedule:
Calcul symbolique-numérique d’intégrales de volumes, séminaire AriC, LIP, ENS Lyon (online).
Périodes : calcul numérique et applications, kick off seminar of ANR project “De rerum natura”, Palaiseau.
Numerical periods in effective algebraic geometry, opening conference of the thematic Einstein semester on algebraic geometry, FU Berlin.
Finding one root of a polynomial system (How to improve the complexity?), Complexity of numerical computation, Felipe's Fest, Berlin.
2020
w/ P. Bürgisser and F. Cucker
In “Rigid continuation paths II”, we develop an algorithm to compute an approximate zero of a polynomial system given as a black-box evaluation function. It follows the main lines of the algorithm from Part I but with a probabilistic estimation of the step length for the numerical continuation.
Going a step further, we show that, on average, the algorithm behaves well on a universal family of polynomial system with low evaluation cost. More precisely, we introduce the model of Gaussian random algebraic branching program and show that our algorithm performs $\operatorname{poly}(n, \delta)$ operations on average to solve a random ABP in $n$ variables, of degree $\delta$ and of size $\operatorname{poly}(n, \delta)$.