Regularity of quasi-symbolic and bracket powers of Borel type ideals

In this paper, we show that the regularity of the q-th quasi-symbolic power $I^{((q))}$ and the regularity of the $q$-th bracket power $I^{[q]}$ of a monomial ideal of Borel type $I$, satisfy the relations $reg(I^{((q))})\leq q \cdot reg(I)$, respectively $reg(I^{[q]})\geq q\cdot reg(I)$. Also, we give an upper bound for $reg(I^{[q]})$.Comment: 8 pages, to appear in Romanian Journal of Mathematics and Computer Scienc

On the Stanley depth of edge ideals of line and cyclic graphs

We prove that the edge ideals of line and cyclic graphs and their quotient rings satisfy the Stanley conjecture. We compute the Stanley depth for the quotient ring of the edge ideal associated to a cycle graph of length $n$, given a precise formula for $n\equiv 0,2 \pmod{3}$ and tight bounds for $n\equiv 1 \pmod{3}$. Also, we give bounds for the Stanley depth of a quotient of two monomial ideals, in combinatorial terms.Comment: 8 pages. Will appear in Romanian Journal of Mathematics and Computer Scienc

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