# 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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