Certified numerical implementation of Zariski-Van Kampen method.

Van Kampen’s motivation for his celebrated theorem was to give a proof of a fact known by Zariski: that the fundamental group of the complement of a complex plane curve is given by its braid monodromy. Moreover, Lefschetz hyperplane theorem allows to reduce to this case the computation of the complement of every hypersurface in the projective (or affine) space. However, there are no purely algebraic methods to compute this braid monodromy. We present a numerical, but certified, method using interval arithmetic and Newton’s interval criterion. The use of interval arithmetic allows us to work with arbitrary coefficients (either rational, algebraic or even transcendental).