Algebraic structure of the range of a trigonometric polynomial

TL;DR

This paper investigates the geometric properties of the range of complex trigonometric polynomials, showing it is mostly contained within a real algebraic set with finite differences, highlighting the role of symmetry.

Contribution

It establishes that the range of a complex trigonometric polynomial is contained in a real algebraic set, with finite differences except in symmetric cases.

Findings

Range is contained in a real algebraic set

Finite difference between the range and the algebraic set

Symmetry affects the containment

Abstract

The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with certain symmetry.