Stefan Kranich (TU Munich)

26 Jan 2017 - W01 0-012 (Wechloy), 4 pm

An epsilon-delta bound for plane algebraic curves

Abstract: Given a plane algebraic curve C: f(x, y) = 0, x_1 ∈ ℂ not a singularity of y w.r.t. x, and ε > 0, we can compute δ > 0 such that |y_j(x_1) - y_j(x_2)| < ε for all holomorphic functions y_j(x) which satisfy f(x, y_j(x)) = 0 in a neighbourhood of x_1 and for all x_2 with |x_1 - x_2| < δ.

We discuss this result and two applications: Firstly, algorithms for reliable homotopy continuation of plane algebraic curves and systems of plane algebraic curves; secondly, a GPU-based algorithm for visualisation of algebraic Riemann surfaces.