The author introduces a family of \(M\)-concave functions arising naturally from polynomials over a field of generalized Puiseux series with prescribed nonvanishing properties. This family contains several \(M\)-concave functions studied in the literature. It is shown that the tropicalization of the space of polynomials with the half-plane property is strictly contained in the space of \(M\)-concave functions. Based on a result of Hardy and Hutchinson a short proof of Speyer’s “hive theorem” is given, which he had used to give a new proof of Horn’s conjecture on eigenvalues of sums of Hermitian matrices. Finally, it is shown that a natural extension of Speyer’s theorem to higher dimensions is false.