Complex solutions of polynomial equations on the unit circle

TL;DR

This paper investigates the properties and decidability of polynomial systems with solutions on the unit circle, connecting algebraic geometry with classical number theory and providing elementary proofs and educational insights.

Contribution

It introduces a criterion for Zariski density of solutions on the unit circle and establishes the decidability of related polynomial systems, contrasting with the Manin-Mumford conjecture.

Findings

Criteria for Zariski density of solutions

Decidability of polynomial systems on the unit circle

Elementary proofs and expository approach

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.