Singular symmetric matrices
Let \(N(T, n, r)= \text{card} \{A\mid A\in S_n (\mathbb{Z})\), \(\text{rank } A=r\), \(\text{tr } A^2< T\}\), where \(S_n (\mathbb{Z})\) denotes the lattice of symmetric \(n\times n\) matrices with entries in \(\mathbb{Z}\), the ring of rational integers. Assuming that \(n\geq 4\) the authors prove that \[ c_1 T^{nr/2}< N(T, n, r)< c_2 T^{nr/2} \qquad \text{for} \quad 3\leq r\leq n-2 \] with \(c_1\), \(c_2\) depending at most on \(n\) and \(r\) (but not on \(T\)), and give an asymptotic formula for \(N(T, n, n- 1)\) as \(T\to \infty\). They cite a recent paper of \textit{W. M. Schmidt} [Acta Arith. 71, 343-375 (1995; Zbl 0784.11055)]\ containing an asymptotic formula for \(N(T, n, 2)\), \(n\geq 3\), \(T\to \infty\). The problem under consideration is reduced to evaluation of the number of certain sublattices of \(\mathbb{Z}^n\), and this is done by employing the methods of geometry of numbers, the early results of \textit{W. M. Schmidt} [Duke Math. J. 35, 327-339 (1968; Zbl 0172.063)]\ being an essential ingredient of the proof.
scientific article; zbMATH DE number 815784
scientific article