A circle and a confocal ellipse have at most ten distance-stationary points

Abstract Let a circle and an ellipse in ℝ 3 have a common focus at the centre of the circle. We prove that the squared distance between a point of the circle and a point of the ellipse has at most ten stationary points, and that ten stationary points do occur. This answers the second question of Problem 8 in the list of Albouy, Cabral and Santos (2012). The question, posed by G. F. Gronchi, asks whether the bound 12 for two confocal ellipses can be lowered to 10 when one of them is a circle. Counting the stationary points amounts to counting the real zeros of a trigonometric polynomial of degree six whose coefficients depend on four parameters, and our proof of the bound is computer-assisted. We subdivide the compactified parameter space into boxes and certify each box in one of two ways: by certified signs of derivatives on arcs of the circle, which bound the number of real zeros directly, or by a winding-number argument that counts the zeros of an associated polynomial inside a contour. The degenerate limits, among them the parabolic limit in which the polynomial acquires twelve real zeros, are treated with exact weighted blow-ups. All floating-point errors are bounded explicitly, and every certificate is recorded, replayed and checked independently. Mathematics Subject Classification (2020): 70F15 · 26C10 · 42A05 · 65G30