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

TL;DR

This paper introduces an epsilon-delta bound for plane algebraic curves that enables reliable homotopy continuation, facilitating the deformation analysis of algebraic systems with potential applications in Darboux transforms.

Contribution

It provides a novel epsilon-delta bound for plane algebraic curves and an algorithm for certified homotopy continuation of such systems.

Findings

The epsilon-delta bound guarantees control over solution variations.

An algorithm for reliable homotopy continuation of plane algebraic curves is developed.

Application demonstrated on continuous deformation of Darboux transforms.

Abstract

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.