2026/07/21 by Benjamin Braun, Christopher O'Neill, Antwon Park
#math.CO #math.AC
Quasi-polynomials are ubiquitous in combinatorics and algebra, as they arise in a variety of enumeration problems. Because quasi-polynomials consist of constituent polynomials, their behavior is more subtle than for a single polynomial. In particular, unlike for a polynomial, it is possible for a quasi-polynomial defined on the positive integers to have infinitely many points at which it is decreasing. In this work, we characterize quasi-polynomials of degree d and period dividing p that are eventually nondecreasing, i.e., that have only finitely many values at which they decrease. We then give a detailed analysis of the space of eventually nondecreasing quasi-polynomials with fixed degree d and period dividing a fixed p such that the h-vector of the quasi-polynomial is nonnegative. Using this analysis, we determine the rate of growth of the number of such quasi-polynomials as a function of the sum of the h-vector entries for the 0-th constituent polynomial.