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

Abstract

We explain how, given a plane algebraic curve $\mathcal{C}:f(x,y)=0$, $x_1\in \mathbb{C}$ not a singularity of $y$ w.r.t. $x$, and $\epsilon >0$, we can compute $\delta >0$ such that $|y_j(x_1)-y_j(x_2)|<\epsilon$ 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.

arXiv commentminor revisions; 20 pages, 1 figure, 2 tables; supplementary video and source code included in arXiv sources

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