Roots of Descent Polynomials and an Algebraic Inequality on Hook Lengths

By reinterpreting the descent polynomial as a function enumerating standard Young tableaux of a ribbon shape, we use Naruse's hook-length formula to express the descent polynomial as a product of two polynomials: one is a trivial part which is a product of linear factors, and the other comes from the excitation factor of Naruse's formula. We expand the excitation factor positively in a Newton basis which arises naturally from Naruse's formula. Under this expansion, each coefficient is the weight of a certain combinatorial object, which we introduce in this paper. We introduce and prove the "Slice and Push Inequality", which compares the weights of such combinatorial objects. As a consequence, we establish a proof of a conjecture by Diaz-Lopez et al. that bounds the roots of descent polynomials.