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

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 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 2π n and 2 sin 2π n , which we denote by Ψn(x, y) and Πn(x, y), respectively. The remaining two forms that we consider are homogenizations of Chebyshev polynomials of first and second kinds, denoted Tn(x, y) and Un(x, y), respectively.

On the Representation of Integers by Binary Forms Defined by Means of the Relation $(x + yi)^n = R_n(x, y) + J_n(x, y)i$

“…Fouvry and Waldschmidt [3] estimated C Φn for cyclotomic binary forms Φ n (x, y). In [5], the author estimated C F for Chebyshev binary forms of the first kind T n (x, y) and of the second kind U n (x, y), as well as for the minimal binary forms Ψ n (x, y) of 2 cos 2π n and Π n (x, y) of 2 sin 2π n . In this, article, we compute C F for two families of binary forms.…”

“…Assume for a contradiction that D 2 Aut |I n |. By[5, Lemma 3.4], there must exist a non-zero rational number t such that either Aut |I n…”