One-dimensional discrete models of maximum likelihood degree one

We settle a conjecture by Bik and Marigliano stating that the degree of a one-dimensional discrete model with rational maximum likelihood estimator is bounded above by a linear function in the size of its support, therefore showing that there are only finitely many fundamental such models for any given number of states. We study these models from a combinatorial perspective with regard to their existence and enumeration. In particular, sharp models, those whose degree attains the maximal bound, enjoy special properties and have been studied as monomial maps between unit spheres. In this way, we present a novel link between Cauchy-Riemann geometry and algebraic statistics.