In this paper we introduce an effective algebraic method for the computation of all the stationary points of the squared distance d2 between a point on one ellipse and a point on a second ellipse with a focus in common with the first one. This problem comes from celestial mechanics, in which the minima between two elliptic Keplerian orbits are relevant to study the probability of collision; some applications of our algorithm in this field are shown. This algorithm is based on the use of the fast Fourier transform to obtain the coefficients of the resultant of the two bivariate components of the gradient of d2 with respect to one variable and relies on specific tools in symbolic computation. An upper bound to the total number of stationary points that we have to expect in this problem is also given; this is done using some tools from algebraic geometry.
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