T. MULDERS

For a real square matrix $M$, Hadamard's inequality gives an upper bound $H$ for the determinant of $M$. This upper bound is sharp if and only if the rows of $M$ are orthogonal. In this paper we study how much we can expect that $H$ overshoots the determinant of $M$, when the rows of $M$ are chosen randomly on the surface of the sphere. This...

In this paper we consider deterministic computation of the exact determinant of a dense matrix $M$ of integers. We present a new algorithm with worst case complexity $O\left(n^4(\log n+\log||M||)+n^3\log^2||M||\right)$, where $n$ is the dimension of the matrix and $||M||$ is a bound on the entries in $M$, but with average expected complexity...