A lower bound of Stanley depth of monomial ideals (Q412530)

Let \(I\) be a monomial ideal in the polynomial ring \(S=K[x_1,\dots,x_n]\) with \(K\) a field. The Stanley depth, \(\text{sdepth}\), of a multigraded module (such as \(I\) or \(S/I\)) was defined by \textit{R. P. Stanley} [Invent. Math. 68, 175--193 (1982; Zbl 0516.10009)]. In the same paper, Stanley posed the conjecture that \(\text{sdepth}(I)\geq\text{depth}(I)\). One of the lines of research around this conjecture proceeds by giving bounds for the Stanley depth in different situations.NEWLINENEWLINEThe main result of this paper (Theorem 2.3) states that for a monomial ideal \(I\) minimally generated by \(m\) monomial generators we have NEWLINE\[NEWLINE\text{sdepth}(I)\geq\max\{ 1,n-\lfloor \frac{m}{2}\rfloor\}NEWLINE\]NEWLINE This improves previous bounds in [\textit{J. Herzog, M. Vladiou} and \textit{X. Zheng}, J. Algebra 322, No. 9, 3151--3169 (2009; Zbl 1186.13019)] and in the squarefree case it answers a question in [\textit{Y. H. Shen}, J. Algebra 321, No. 4, 1285--1292 (2009; Zbl 1167.13010)].