The unimodality of pure $$O$$-sequences of type two in four variables.(English)Zbl 1332.05157

Summary: Since the 1970’s, great interest has been taken in the study of pure $$O$$-sequences, which, due to Macaulay’s theory of inverse systems, have a bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these according to their shape. Macaulay’s theorem immediately gives us that all Artinian algebras in two variables have unimodal Hilbert functions. Furthermore, it has been shown that all Artinian level monomial algebras of type two in three variables have unimodal Hilbert functions. This paper will classify all Artinian level monomial algebras of type two in four variables into two classes of ideals, prove that they are strictly unimodal and show that one of the classes is licci.

MSC:

 05E40 Combinatorial aspects of commutative algebra 13C40 Linkage, complete intersections and determinantal ideals 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13E10 Commutative Artinian rings and modules, finite-dimensional algebras 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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