Complex solutions of polynomial equations on the unit circle

Abstract

We explore systems of polynomial equations where we seek complex solutions with absolute value 1. Geometrically, this amounts to understanding intersections of algebraic varieties with tori -- Cartesian powers of the unit circle. We study the properties of varieties in which this intersection is Zariski dense, give a criterion for Zariski density and use it to show that the problem is decidable. This problem is a ``continuous'' analogue of the Manin-Mumford conjecture for the multiplicative group of complex numbers, however, the results are very different from Manin-Mumford. While the results of the paper appear to be new, the proofs are quite elementary. This is an expository article aiming to introduce some classical mathematical topics to a general audience. We also list some exercises and problems at the end for the curious reader to further explore these topics.