An upper bound on the number of classes of perfect unary forms in totally real number fields (Q6591606)

Let \(K\) be a totally real number field of degree \(r\) and let \(\omega_1, \ldots, \omega_r\) be a basis in the ring of algebraic integers \(\mathcal{O}_K\) in \(K\). For each totally positive \(a\in K\) (denoted by \(a\in K_{>>0}\)) one can associate the positive definite rational quadratic form\N\[\N\mathrm{Tr}_{K/\mathbb{Q}} ax^2 = \mathrm{Tr}_{K/\mathbb{Q}} a(x_1\omega_1+\ldots + x_r\omega_r)^2\N\]\Nof \(m\) variables. Denote by \(\mu(a)=\min \{\mathrm{Tr}_{K/\mathbb{Q}} av^2 \ | \ 0\ne v\in \mathcal{O}_K \}\) and \(\mathcal{M}(a)= \{v\in \mathcal{O}_K \ | \ \mathrm{Tr}_{K/\mathbb{Q}} av^2 = \mu(a)\}\) the minimum and the set of minimal vectors respectively. A unary quadratic form \(ax^2\) (\(a\in K\) is totally positive) is called perfect, if the system of linear equations \(\mathrm{Tr}_{K/\mathbb{Q}} av^2 = \mu(a)\), \(v\in \mathcal{M}(a)\), yields a unique solution \(a\) (see [\textit{M. Koecher}, Math. Ann. 141, 384--432 (1960; Zbl 0095.25301)]).\N\NUsing the volumetric argument (see [\textit{W. P. J. van Woerden}, Adv. Math. 365, Article ID 107031, 12 p. (2020; Zbl 1444.11150)]) the authors obtained an upper bound for the number of unary perfect forms (up to equivalence and scaling) (see Theorem 1). This upper bound depends on the covering radius of the log unit lattice in the infinity norm and additive Hermite-Humbert number\N\[\N\gamma_K=\sup_{a\in K_{>>0} } \frac{\mu(a)}{\mathrm{Nm}_{K/\mathbb{Q}}(a)^{1/[K\colon \mathbb{Q}]}}.\N\]\NMoreover, the obtained upper bound was simplified under the assumption that \(K\) is a Galois extension of prime degree (Theorem 2) or \(K\) is a maximal totally real subfield of a cyclotomic field (Theorem 3). In both cases neither covering radius nor \(\gamma_K\) were used.