Equivalence of the four notions of a branch at infinity

Prove that the four proposed descriptions of a branch of an irreducible affine curve at a point at infinity are equivalent: the Puiseux-series description, the local analytic irreducible-component description, the minimal-prime description in the completed local ring, and the description via points in the normalization lying over the point at infinity.

Background

The appendix proposes four interpretations of branches of an irreducible curve at infinity. The first uses Puiseux-series equations in local coordinates; the second uses local analytic irreducible components; the third uses minimal prime ideals in the completed local ring; and the fourth uses points in the normalization lying over the point at infinity.

The authors explicitly state that they believe these interpretations are equivalent but do not provide a rigorous proof. They also note technical subtleties in assembling coordinatewise Puiseux expansions and in establishing the correspondence between analytic and completed-local-ring components. Thus, a rigorous equivalence theorem remains unresolved in the paper.