On the Amick-Fraenkel conjecture

In 1987, C. J. Amick and L. E. Fraenkel published a paper on the behaviour of the Stokes wave of extreme form near its crest, where they obtained a complete asymptotic expansion for the angle between the wave profile and the horizontal. Their derivation of the expansion relied on an assumption of linear independence over the rationals of solutions of a certain transcendental equation, which is still an open question. We show that Schanuel's conjecture implies not only the conjectured linear independence of solutions, but also their algebraic independence.