Let I be a homogeneous ideal in polynomial ring over field. $$I^{(n)}$$ the n-th symbolic power of I. Motivated by results about ordinary powers I, we study asymptotic behavior regularity function $${{\,\mathrm{reg}\,}}(I^{(n)})$$ and maximal generating degree $$\omega (I^{(n)})$$ , when is monomial ideal. It known that both functions are eventually quasi-linear. We show that, addition, sequenc...

Let I be a matroidal ideal of degree d polynomial ring $$R=K[x_1,\ldots ,x_n]$$ , where K is field. $${\text {astab}}(I)$$ and {dstab}}(I)$$ the smallest integers m n, for which {Ass}}(I^m)$$ {depth}}(I^n)$$ stabilize, respectively. In this paper, we show that {astab}}(I)=1$$ if only {dstab}}(I)=1$$ . Moreover, prove $$d=3$$ then {astab}}(I)={\text Furthermore, an almost square-free Veronese ty...