In the present paper, we study the Dao numbers $\mathfrak{ d}_1(I),\mathfrak{ d}_2(I)$ and $\mathfrak{ d}_3(I)$ of an ideal $I$ of a Noetherian local ring $(R,\mathfrak{ m},K)$ or a standard graded Noetherian $K$-algebra. They are defined as the smallest $\ell\ge0$ such that $I\mathfrak{ m}^k$ is $\mathfrak{ m}$-full, full, weakly $\mathfrak{ m}$-full, respectively, for all $k\ge\ell$. We provide general bounds for the Dao numbers in terms of the Castelnuovo-Mumford regularity of certain modules over the Rees algebra $\mathcal{ R}(\mathfrak{ m})$. If $R$ is a Koszul algebra, we prove that the Dao numbers are less or equal to $\text{ reg}_{ \text{ gr}_\mathfrak{ m}(R)}\text{ gr}_\mathfrak{ m}(I)$, where $\text{ gr}_\mathfrak{ m}(I)$ is the associated graded module of $I$. Finally, for monomial ideals, we combinatorially bound the Dao numbers in terms of asymptotic linear quotients and bounding multidegrees.
Categories: math.AC math.CO