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Regularity of Powers of Quadratic Sequences with applications to binomial edge ideals

Research paper by A. V. Jayanthan, Arvind Kumar, Rajib Sarkar

Indexed on: 07 Oct '19Published on: 04 Oct '19Published in: arXiv - Mathematics - Commutative Algebra



Abstract

In this article we obtain an upper bound for the regularity of powers of ideals generated by a homogeneous quadratic sequence in a polynomial ring in terms of regularity of its related ideals and degrees of its generators. As a consequence we compute upper bounds for $reg(J_G^s)$ for some classes of graphs $G$. We generalize a result of Matsuda and Murai to show that the Castelnuovo-Mumford regularity of $J^s_G$ is bounded below by $2s+\ell(G)-1$, where $\ell(G)$ is the longest induced path in any graph $G$. Using these two bounds we compute explicitly the regularity of powers of binomial edge ideals of cycle graphs, star graphs and balloon graphs. Also we give a sharp upper bound for the regularity of powers of almost complete intersection binomial edge ideals.