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.


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


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