When is a trigonometric polynomial not a trigonometric polynomial?

TL;DR

This paper explores the distinction between different notions of trigonometric polynomials using algebraic geometry, revealing that the standard definition differs from a more naive interpretation.

Contribution

It demonstrates, via Bézout's theorem, that the standard and naive notions of trigonometric polynomials are not equivalent.

Findings

Standard and naive notions of trigonometric polynomials differ.

Algebraic geometry provides tools to distinguish these notions.

Bézout's theorem is applied to analyze trigonometric polynomial definitions.

Abstract

As an application of B\'ezout's theorem from algebraic geometry, we show that the standard notion of a trigonometric polynomial does not agree with a more naive, but reasonable notion of trigonometric polynomial.