A special class of matrices (Q1131285)

The subject of this paper has its origins in a conjecture of Minkowski in the classical geometry of numbers. This says, essentially, that the critical lattices of the \(n\)-dimensional cube \(| x_1|\leq 1, \ldots, | x_n|\leq 1\) are given by linear forms with triangular matrices and ones in the principal diagonal. The conjecture was finally proved by \textit{G. Hajós} [Math. Z. 47, 427--467 (1941; Zbl 0025.25401)], but \textit{C. L. Siegel} [Math. Ann. 87, 36--38 (1922; JFM 48.0166.02)] had previously tried to prove it by showing that each point \(\neq 0\) of any critical lattice has at least one coordinate in the set of non-zero integers. The present paper arises out of an attempt to generalize Siegel's ideas to lattices in algebraic number fields. Let \(D\) be an integral domain, \(K\) its quotient field, \(D^n\) the set of all \(n\)-by-\(1\) matrices over \(D\) and \(A\) an \(n\)-by-\(n\) matrix over a field containing \(K\). The set of points \(Au\), \(u\in D^n\), is obviously the analogue of a lattice in the classical case. A matrix \(A\) is said to have property \((P_D)\) if and only if for all \(u\neq 0\) in \(D^n\) the vector \(Au\) has at least one component in \(D^* = D-\{0\}\). If \(P\) is a permutation matrix, \(T\) is a lower triangular matrix with only ones in the principal diagonal and \(N\) is non-singular over \(D\), then \(A = PTN\) has property \((P_D)\). The authors have proved elsewhere [Bull. Am. Math. Soc. 66, 118--123 (1960; Zbl 0093.05001)] that for \(D=Z\) there are matrices not of the form \(PTN\) which have property \((P_D)\). In the same paper they proved that if \(D\) is the ring of integers of an algebraic number field of class number 1, a necessary condition for \(A\) to have \((P_D)\) is that \(\det A\in D^*\). In this paper they prove the following extensions of this theorem. Theorem 1. Suppose that \(K\) is algebraic over the prime field therein, and that \(A\) is an \(n\)-by-\(n\) matrix, with \(n\leq\#(K)\) \((\#\) denotes cardinal number). Then: (i) if \(K\) is of prime characteristic, \(A\) has property \((P_D)\) if and only if \(A =PTN\); (ii) if \(D\) is a Dedekind domain and \(K\) is a finite algebraic extension of the rationals, for \(A\) to have \((P_D)\) we must have \(\det A\in D^*\). Theorem 2. If \(D=D_1[t]\), where \(t\) is transcendental over \(D_1\), if \(\# (D_1) > n\), and if \(A\) has \((P_D)\) then the rows of \(A\) can be so ordered that the matrices \(A_r\) of the first \(r\) rows of \(A\) have all \(r\)-by-\(r\) minors in \(D\) and not all zero, for \(r = 1, 2, \ldots, n\). In particular, the first row is over \(D\), and \(\det A\in D^*\). In the case when \(D_1[t]\) is a Gaussian domain, Theorem 2 can be improved to an ``if and only if'' statement. The proofs of these theorems depend mainly on matrix calculation and, at one point, on properties of discrete valuation rings.