Estimates of Hermite constants for algebraic number fields (Q2769542)

Consider a field \(k\) of degree \(r\) (conjugates \(k_j\)) and a variable set \(g\) of \(r\) unimodular \(n\times n\) real positive definite matrices \(g_j\) (where \(g_m= g_p\) for complex-conjugate \(k_m,k_p\)). Then a variable nonzero \(n\)-vector \(\mu\) of \(k\)-integers produces quadratic norms \(g_j(\mu_j)\) arising for each of the \(r\) conjugates \(g_j\) acting on the \(n\)-vector \(\mu_j\). This defines a norm \(N(g,\mu)= \prod g_j(\mu_j)\) and the extended Hermite constants are defined as \(\gamma_n(k)= \max_g \min_\mu N(g,\mu)\) [see \textit{M. I. Icaza}, J. Lond. Math. Soc. (2) 55, 11-22 (1997; Zbl 0874.11047) for a simple presentation; an adelic version appears in \textit{J. L. Thunder}, Mich. Math. J. 45, 301-314 (1998; Zbl 1007.11044)]. NEWLINENEWLINENEWLINEFor \(k= \mathbb{Q}(r=1)\), familiarly, \(\gamma_2= \sqrt{4/3},\dots, \gamma_8=2\), etc. For totally real \(k\), these constants come into the floor of the fundamental domain of the Hilbert modular group [see the reviewer, Proc. Symp. Pure Math. 8, 190-202 (1965; Zbl 0137.05702); Math. Comput. 19, 594-605 (1965; Zbl 0144.28501); J. Math. Anal. Appl. 15, 55-59 (1966; Zbl 0146.11101)]. The class number of \(k\) is seen to enter the calculations of \(\gamma_n(k)\). NEWLINENEWLINENEWLINEIn the present work, the authors improve the upper and lower estimates for \(\gamma_n(k)\). The theorems are much too recondite to even summarize here, but numerical work for fields of degree 4 to 8 is offered in support.