Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 the maximal ideal $\frak{m}=(x_1,...,x_n)$.
  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:
  \begin{itemize} \item[(i)] $I$ is a matroidal ideal and $n\leq 5$.
\item[(ii)] $I$ is a polymatroidal ideal, $n=4$ and
$\frak{m}\notin\Ass^{\infty}(I)$, where $\Ass^{\infty}(I)$ is the stable set of
associated prime ideals of $I$. \item[(iii)] $I$ is a polymatroidal ideal of
degree $2$. \end{itemize} Moreover, we give an example of a polymatroidal ideal
for which $\astab(I)\neq\dstab(I)$. This is a counterexample to the conjecture
of Herzog and Qureshi, according to which these two numbers are the same for
polymatroidal ideals.

30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg