  In this article, we investigate the quantitative unique continuation
properties of real-valued solutions to elliptic equations in the plane. Under a
general set of assumptions on the operator, we establish quantitative forms of
Landis' conjecture. Of note, we prove a version of Landis' conjecture for
solutions to $-\Delta u + V u = 0$, where $V$ is a bounded function whose
negative part exhibits polynomial decay at infinity. The main mechanism behind
the proofs is an order of vanishing estimate in combination with an iteration
scheme. To prove the order of vanishing result, we present a new idea for
constructing positive multipliers and use it reduce the equation to a Beltrami
system. The resulting first-order equation is analyzed using the similarity
principle and the Hadamard three-quasi-circle theorem.
