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Journal of the Ramanujan Mathematical Society

Volume 28, Issue 2, June 2013  pp. 195–212.

On the periodicity of the first betti number of the semigroup rings under translations

Authors Adriano Marzullo
Author institution: Department of Mathematics, Becker College, Worcester Massachusetts, 01609

Summary:  Let k be a field of characteristic zero. Given an ordered 3-tuple of positive integers a = (a, b, c) and for, a family of sequences, we consider the collection of monomial curves in associated with a. The Betti numbers of the Semigroup rings collection associated with a are conjectured to be eventually periodic with period a + b + c by Herzog and Srinivasan. Let, in this paper, we prove that for a = (p(b + c), b, c) or a = (a, b, p(a + b)) in the collection of defining ideals associated with, for large j the ideals are complete intersections if and only if (a + b + c)|j. Moreover, the complete intersections are periodic with the conjectured period.

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