On-line fault-detection in uncertain measurement and estimation systems is of particular interest in many applications. In certain systems based on the Kalman filter, this test can be performed by checking whether hyperellipsoids overlap. This test can be applied to detecting failure in the system itself or in the sensors used to determine the system state. To facilitate the practical application of such tests, we describe a simple condition for overlap of two ellipsoids and propose an efficient algorithmic implementation for testing this condition. There are applications in many other areas, such as collision avoidance or computer graphics. Our proposal makes use of Leverriere’s algorithm and Sturm’s theorem, a result of algebraic geometry. Thus, no approximative methods, such as root finding or minimization are needed. Furthermore, the complexity of the algorithm is fixed for a fixed problem dimension.