On Convexity of Polynomial Paths and Generalized Majorizations

Abstract

In this paper we give some useful combinatorial properties of polynomial paths. We also introduce generalized majorization between three sequences of integers and explore its combinatorics. In addition, we give a new, simple, purely polynomial proof of the convexity lemma of E. M. de Sá and R. C. Thompson. All these results have applications in matrix completion theory.