On the similarity group of forms of higher degree
Let \((V, \theta)\) be an at least 1-regular \(d\)-linear space over \(k\), \(\text{char} (k) = 0\) or \(\text{char} (k) > d\). A linear map \(\sigma \in \Hom (V,V)\) is called a similarity of \((V, \theta)\) with norm \(\lambda_ \sigma \in k^ \times\) if \(\theta (\sigma (v_ 1, \dots, v_ d)) = \lambda_ \sigma \theta (v_ 1, \dots, v_ d)\) \(\forall v_ 1, \dots, v_ d \in V\). The similarities of a nonsingular space form a group, called similarity group \(S_ k (\theta)\). The subgroup of automorphisms is called \(\Aut_ k (\theta)\). Using Kummer's theory of fields the author proves one of the main results of this paper. Theorem: \(S_ k (\theta)/ \Aut_ k (\theta) k^ \times\) is a finite abelian group of exponent \(d\). Applying this result he completely characterizes the similarity group of a binary form of odd degree over \(\mathbb{Q}\). Distinguishing between the nonsingular and the singular case he gets the following results: Theorem: Let \(f\) be a nonsingular binary form of odd degree \(d\) over \(k = \mathbb{Q} (\zeta)\) with \(\zeta\) a primitive \(d\)-th root of 1. Then \(S_ k(f) = \Aut_ k (f) \times k^ \times / \Delta_ d (k)\), where \(\Delta_ d (k)\) is the image of \(\mu_ d (k)\) under \(\eta \mapsto (m_ \eta, \eta^{-1})\). Theorem: Let \(f(X,Y)\) be a form of odd degree \(d \geq e\) over \(\mathbb{Q}\). If \(f \not \cong \alpha X^ e\) \(Y^{d-e}\), then \(S_ Q(f) \cong \Aut_ \mathbb{Q} (f) \times \mathbb{Q}^ \times\).
scientific article; zbMATH DE number 645117