Regularity of powers of quadratic sequences with applications to binomial ideals

Abstract

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

17 pages. Example 2.9 has been added and section about LSS ideals has been added

References in corpus (6)

• Regularity and cohomology of determinantal thickenings
• Parity binomial edge ideals
• Upper bounds for the regularity of powers of edge ideals of graphs
• Regularity of binomial edge ideals of chordal graphs
• Binomial Edge Ideals of Generalized block graphs
• Almost complete intersection binomial edge ideals and their Rees algebras

Cited by in corpus (8)

• Powers of binomial edge ideals with quadratic Gröbner bases
• Cohen-Macaulay binomial edge ideals and accessible graphs
• D-Sequence Edge Binomials, and Regularity of Powers of Binomial Edge Ideals of Trees
• Regularity of powers of d-sequence (parity) binomial edge ideals of unicycle graphs
• Comparison of symbolic and ordinary powers of parity binomial edge ideals
• (Almost) Complete Intersection Lovász-Saks-Schrijver ideals and regularity of their powers
• Powers of facet ideals of simplicial trees
• Algebraic properties of binomial edge ideals of Levi graphs associated with curve arrangements