S-curves in Polynomial External Fields

Curves in the complex plane that satisfy the S-property were first introduced by Stahl and they were further studied by Gonchar and Rakhmanov in the 1980s. Rakhmanov recently showed the existence of curves with the S-property in a harmonic external field by means of a max-min variational problem in logarithmic potential theory. This is done in a fairly general setting, which however does not include the important special case of an external field given by the real part of a polynomial of degree greater than or equal to 2. In this paper we give a detailed proof of the existence of a curve with the S-property in the external field given by the real part of a polynomial V, within the collection of all curves that connect two or more pre-assigned directions at infinity in which the real part of V grows. Our method of proof is very much based on the works of Rakhmanov on the max-min variational problem and of Mart\'inez-Finkelshtein and Rakhmanov on critical measures.