On the Area of the Fundamental Region of a Binary Form Associated with Algebraic Trigonometric Quantities

TL;DR

This paper estimates the area enclosed by the level curve |F(x, y)|=1 for specific families of binary forms linked to algebraic trigonometric quantities, including minimal polynomials and Chebyshev polynomials.

Contribution

It provides explicit area estimates for the fundamental regions of four families of binary forms associated with algebraic trigonometric quantities.

Findings

Derived bounds for the area of the fundamental region for each family.

Established connections between algebraic trigonometric quantities and binary form geometry.

Extended previous work on binary forms to include forms related to cosine and sine minimal polynomials.

Abstract

Let $F(x, y)$ be a binary form of degree at least three and non-zero discriminant. We estimate the area $A_F$ bounded by the curve $|F(x, y)| = 1$ for four families of binary forms. The first two families that we are interested in are homogenizations of minimal polynomials of $2\cos\left(\frac{2\pi}{n}\right)$ and $2\sin\left(\frac{2\pi}{n}\right)$, which we denote by $\Psi_n(x, y)$ and $\Pi_n(x, y)$, respectively. The remaining two families of binary forms that we consider are homogenizations of Chebyshev polynomials of first and second kinds, denoted $T_n(x, y)$ and $U_n(x, y)$, respectively.