Estimates on conjectures of Minkowski and Woods
Let \(L_i=a_{i1}x_1 + \dots + a_{in}x_n\), \(1\leq i \leq n\), be \(n\) real linear forms in \(n\) real variables with determinant \(\Delta=\det(a_{ij}) \neq 0\). Let \(c_1, \ldots, c_n\) be real numbers. \textit{H. Minkowski} is believed to have conjectured that there exist integers \(u_1, \ldots, u_n\) such that \[ \prod_{i=1}^n |L_i(u_1, \ldots, u_n)+c_i| \leq \frac{1}{2^n} |\Delta|. \] He proved this for \(n=2\), [Math. Ann. 54, 91--124 (1901; JFM 31.0213.02)] and it has now been verified for \(2\leq n\leq 8\). Define \[ M_n = \sup_{L_1, \ldots, L_n} \sup_{(c_1, \ldots, c_n) \in \mathbb R^n} \inf_{(u_1, \ldots, u_n) \in \mathbb Z^n} \prod_{1\leq i \leq n} |L_i(u_1,\ldots, u_n)+c_i|. \] Minkowski's Conjecture is equivalent to \(M_n \leq \frac{1}{2^n} |\Delta|\). \textit{N. Chebotarev} [Vierteljahresschr. Naturforsch. Ges. 85, Beibl. Nr. 32, 27--30 (Rudolf Fueter Festschr.) (1940; Zbl 0023.20701 and JFM 66.0185.01)] proved the weaker inequality \(M_n \leq \frac{1}{2^{n/2}} |\Delta|\). Several authors, including \textit{I. V. Il'in} [Algebraic systems. Ivanovo: Ivanov. Gos. Univ., 115--125 (1991; Zbl 0956.00528)] have given improved bounds for \(M_n\) of the form \(M_n \leq \frac{1}{\nu_n 2^{n/2}} |\Delta|\). In this paper, the authors give bounds for \(\nu_n\) that improve Il'in's bounds in the range \(9\leq n \leq 22\). For example, Il'in had \(\nu_9=3.315\ldots\); the authors improve this to \(\nu_9=19.396\ldots\). The authors also consider a related conjecture due to \textit{A. C. Woods} [Mathematika 12, 143--150 (1965; Zbl 0135.10701)]. Woods's conjecture is also known to be true for \(n\leq 8\). In this paper, the authors obtain an approximation to Wood's conjecture that is valid for \(n\geq 9\), and they use this to obtain the above mentioned improvements for \(\nu_n\).
scientific article; zbMATH DE number 5794230
scientific article