On the Automorphism Group of a Binary Form Associated with Algebraic Trigonometric Quantities

TL;DR

This paper determines the automorphism groups of specific binary forms linked to algebraic trigonometric quantities, including those derived from minimal polynomials of cosine and sine values, as well as Chebyshev polynomial homogenizations.

Contribution

It computes the automorphism groups for four families of binary forms associated with algebraic trigonometric quantities, a novel analysis in this context.

Findings

Automorphism groups of forms related to cosines and sines are explicitly computed.

Automorphism groups of Chebyshev polynomial forms are characterized.

Results connect algebraic trigonometry with binary form symmetries.

Abstract

Let $F(x, y)$ be a binary form of degree at least three and non-zero discriminant. In this article we compute the automorphism group $\operatorname{Aut} F$ 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 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.