On stability properties of powers of polymatroidal ideals

Let R = K[x(1), ..., x(n)] be the polynomial ring in n variables over a field K with themaximal ideal m = (x(1), ..., x(n)). Let astab(I) and dstab(I) be the smallest integer n for which Ass(I n) and depth(I n) stabilize, respectively. In this paper we show that astab(I) = dstab(I) in the following cases: (i) I is a matroidal ideal and n <= 5. (ii) I is a polymatroidal ideal, n = 4 and m is not an element of Ass(infinity) (I), where Ass(infinity) (I) is the stable set of associated prime ideals of I. (iii) I is a polymatroidal ideal of degree 2. Moreover, we give an example of a polymatroidal ideal for which astab(I) not equal dstab(I). This is a counterexample to the conjecture of Herzog and Qureshi, according to which these two numbers are the same for polymatroidal ideals.