Abstract
In this article we study the Hilbert function HR of one-dimensional semigroup rings R = k[[S]], with embedding dimension four over an infinite field k. Let S = and let M = S ∖{0}. Consider the Apery set of S with respect to the multiplicity e and its subsets Ah = {s ∈ Apery(S) | s ∈ hM ∖ (h + 1)M}, h ≥ 2. Further let D2 ⊆{n3, n4} be the set of generators with torsion order 1. We prove that HR is non-decreasing at level ≤ 3 and that HR is non decreasing in each of the following cases: if A2 has cardinality ≤ 4, if A3 has cardinality ≤ 3, if A4 = ∅, if D2 has cardinality 2, if S has multiplicity ≤ 13.
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