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Masuti","submitted_at":"2017-01-11T22:26:59Z","abstract_excerpt":"In this paper we study the O-sequences of the local (or graded) $K$-algebras of socle degree $4.$ More precisely, we prove that an O-sequence $h=(1, 3, h_2, h_3, h_4)$, where $h_4 \\geq 2,$ is the $h$-vector of a local level $K$-algebra if and only if $h_3\\leq 3 h_4.$ We also prove that $h=(1, 3, h_2, h_3, 1)$ is the $h$-vector of a local Gorenstein $K$-algebra if and only if $h_3 \\leq 3$ and $h_2 \\leq \\binom{h_3+1}{2}+(3-h_3).$ In each of these cases we give an effective method to construct a local level $K$-algebra with a given $h$-vector. 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