Classification of the relative positions between an ellipsoid and an elliptic paraboloid

We classify all the relative positions between an ellipsoid and an elliptic paraboloid when the ellipsoid is small in comparison with the paraboloid ({\it small} meaning that the ellipsoid cannot be tangent to the paraboloid at two points simultaneously). This provides an easy way to detect contact between the two surfaces by a direct analysis of the coefficients of a fourth degree polynomial.