Unmixedness and arithmetic properties of matroidal ideals

Let $$R=k[x_1,\ldots ,x_n]$$R=k[x1,…,xn] be the polynomial ring in n variables over a field k and let I be a matroidal ideal of degree d. In this paper, we study the unmixedness properties and the arithmetical rank of I. Moreover, we show that $$ara(I)=n-d+1$$ara(I)=n-d+1. This answers the conjecture made by Chiang-Hsieh (Comm Algebra 38:944–952, 2010, Conjecture).

The arithmetical rank of the edge ideals of graphs with pairwise disjoint cycles

We prove that, for the edge ideal of a graph whose cycles are pairwise vertex-disjoint, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes.

On the binomial arithmetical rank of toric ideals

In this paper, we investigate sufficient conditions which ensure that the minimal number of generators of a toric ideal IA is equal to its binomial arithmetical rank. We study particularly the case where IA is the toric ideal of a graph.