## Stanley depth of factors of polymatroidal ideals and the edge ideal of forests.(English)Zbl 1354.13017

Let $$S=k[x_1,\dots,x_n]$$ be the polynomial ring in $$n$$ variables over a field $$k$$ and $$J\subseteq I\subseteq S$$ be two polymatroidal ideals. By using the affine ranks of $$I$$ and $$J$$, the authors provide a lower bound for the Stanley depth of $$I/J$$. The affine rank is also used for proving that $$I^k/I^{k+1}$$ satisfies the Stanley’s inequality, that is $$\mathrm{sdepth}(I^k/I^{k+1})\geq\mathrm{depth}(I^k/I^{k+1})$$, for $$k\gg0$$. Moreover they show that $$I^k/I^{k+1}$$, $$k\gg0$$, satisfies the Stanley’s inequality, when $$I$$ is the edge ideals of a forest graph with $$p$$ connected components.

### MSC:

 13C15 Dimension theory, depth, related commutative rings (catenary, etc.) 05E99 Algebraic combinatorics 13C13 Other special types of modules and ideals in commutative rings
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### References:

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