Smooth counterexamples to strong unique continuation for a Beltrami system in $\mathbb{C}^2$

TL;DR

This paper constructs a smooth counterexample in complex analysis showing that strong unique continuation fails for a vector-valued Beltrami system in a0a0complex space, challenging previous assumptions about such systems.

Contribution

It provides the first explicit smooth counterexample to strong unique continuation for a vector Beltrami system in a0a0complex space.

Findings

Counterexample shows failure of strong unique continuation.

The constructed map vanishes to infinite order at the origin.

Ratio of derivatives also vanishes to infinite order.

Abstract

We construct an example of a smooth map $\mathbb{C}\to\mathbb{C}^2$ which vanishes to infinite order at the origin, and such that the ratio of the norm of the $\bar z$ derivative to the norm of the $z$ derivative also vanishes to infinite order. This gives a counterexample to strong unique continuation for a vector valued analogue of the Beltrami equation.