A direct method for checking overlap of two hyperellipsoids

In this work, we propose a method for checking whether two arbitrary-dimensional hyperellipsoids overlap without making use of any optimization or root-finding methods. This is achieved by formulating an overlap condition as a polynomial root counting problem, which can be solved directly. The addressed challenges involve the inversion of a polynomial matrix using a direct method. The proposed approach extends one of our earlier results, which was restricted to certain combinations of ellipsoids and yields a fixed run-time for a fixed problem dimensionality. Thus, for the first time, an algorithm for checking overlap of arbitrary hyperellipsoids is proposed that can be evaluated in closed form. That is, in the absence of cut-off errors, the proposed method yields an exact result after a finite number of steps.