An epsilon-delta bound for plane algebraic curves and its use for certified homotopy continuation of systems of plane algebraic curves

We explain how, given a plane algebraic curve $\mathcal{C}\colon f(x,y) = 0$, $x_1 \in \mathbb{C}$ not a singularity of $y$ w.r.t. $x$, and $\varepsilon > 0$, we can compute $\delta > 0$ such that $|y_j(x_1) - y_j(x_2)| < \varepsilon$ 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 $|x_1 - x_2| < \delta$. Consequently, we obtain an algorithm for reliable homotopy continuation of plane algebraic curves. As an example application, we study continuous deformation of closed discrete Darboux transforms. Moreover, we discuss a scheme for reliable homotopy continuation of triangular polynomial systems. A general implementation has remained elusive so far. However, the epsilon-delta bound enables us to handle the special case of systems of plane algebraic curves. The bound helps us to determine a feasible step size and paths, which are equivalent w.r.t. analytic continuation to the actual paths of the variables but along which we can proceed more easily.