Pure O-sequences in four variables are unimodal through socle degree six

A pure O-sequence is the h-vector of an Artinian monomial level algebra. Equivalently, it counts, degree by degree, the monomials that divide at least one member of a fixed set of monomials of degree e, the socle degree. Migliore, Nagel and Zanello (2012) asked, for a fixed number r of variables, for the largest socle degree e such that every pure O-sequence is unimodal (Problem 3 of their survey). For r = 4 the known bounds were 4 ≤ e ≤ 6: Boyle (2012) proved unimodality through socle degree 4 and constructed non-unimodal examples in every socle degree e ≥ 7. This paper settles the two remaining cases. Every pure O-sequence in four variables of socle degree at most 6 is unimodal, so the answer to Problem 3 for r = 4 is exactly 6. In addition, every pure O-sequence in five variables of socle degree 4 is unimodal. Hence the smallest number of variables admitting a non-unimodal pure O-sequence of socle degree 4 (Problem 4 of the survey) is at least 6. The proofs are computer-assisted. Socle degree 5 in four variables, and socle degree 4 in five variables, are settled by checking all 2^35 candidate layers in degree e − 1. Socle degree 6 is reduced to two extremal quantities of the degree-5 layer, which are computed exactly by dynamic programmes over a splitting by one variable. The bounds used in both proofs are attained, by the pure O-sequences (1, 4, 8, 9, 10, 11) and (1, 4, 10, 17, 19, 19, 20).