  We present a purely algebraic formulation (i.e. polynomial equations only) of
the minimum-cost multi-impulse orbit transfer problem without time constraints,
while keeping all the variables with a precise physical meaning. We apply
general algebraic techniques to solve these equations (resultants, Gr\"obner
bases, etc.) in several situations of practical interest of different degrees
of generality. For instance, we provide a proof of the optimality of the
Hohmann transfer for the minimum fuel 2-impulse circular to circular orbit
transfer problem, and we provide a general formula for the optimal 2-impulse
in-plane transfer between two rotated elliptical orbits under a mild symmetry
assumption on the two points where the impulses are applied (which we
conjecture that can be removed).
