Some remarks on the Stanley's depth for multigraded modules

We show that the Stanley's conjecture holds for any multigraded $S$-module $M$ with \sdepth(M)=0, where $S=K[x_1,...,x_n]$. Also, we give some bounds for the Stanley depth of the powers of the maximal irrelevant ideal in $S$.Comment: 6 page

Stanley depth of monomial ideals with small number of generators

For a monomial ideal $I\subset S=K[x_1,...,x_n]$, we show that \sdepth(S/I)\geq n-g(I), where $g(I)$ is the number of the minimal monomial generators of $I$. If $I=vI'$, where $v\in S$ is a monomial, then we see that \sdepth(S/I)=\sdepth(S/I'). We prove that if $I$ is a monomial ideal $I\subset S$ minimally generated by three monomials, then $I$ and $S/I$ satisfy the Stanley conjecture. Given a saturated monomial ideal $I\subset K[x_1,x_2,x_3]$ we show that \sdepth(I)=2. As a consequence, \sdepth(I)\geq \sdepth(K[x_1,x_2,x_3]/I)+1 for any monomial ideal in $I\subset K[x_1,x_2,x_3]$.Comment: 7 pages. submitted to Central European Journal of Mathematic