let $r=k[x_1,x_2,cdots, x_n]$ be a polynomial ring over a field $k$. we prove that for any positive integers $m, n$, $text{reg}(i^mj^nk)leq mtext{reg}(i)+ntext{reg}(j)+text{reg}(k)$ if $i, j, ksubseteq r$ are three monomial complete intersections ($i$, $j$, $k$ are not necessarily proper ideals of the polynomial ring $r$), and $i, j$ are of the form $(x_{i_1}^{a_1}, x_{i_2}^{a_2}, cdots, x_{i_l...

An equigenerated monomial ideal [Formula: see text] in the polynomial ring is a Freiman if text], where analytic spread of and least number generators text]. In this paper, we classify certain classes Borel-type ideals, including Borel ideals with multiple principal text]-Borel which are Freiman.

We introduce the notion of Q-Borel ideals: ideals which are closed under the Borel moves arising from a poset Q. We study decompositions and homological properties of these ideals, and offer evidence that they interpolate between Borel ideals and arbitrary monomial ideals.

We study ideals of Borel type, including k-Borel and t-spread Veronese ideals. determine their free resolutions homological shift The multiplicity the analytic spread equigenerated squarefree principal are computed. For multiplicity, result is given under an additional assumption which always satisfied for These results used to analyze behavior height, HSj(I) as functions j, when I ideal.

We study certain types of ideals in the standard Borel subalgebra of an untwisted affine Lie algebra. We classify these ideals in terms of the root combinatorics and give an explicit formula for the number of such ideals in type A. The formula involves various aspects of combinatorics of Dyck paths and leads to a new interesting integral sequence.

Strongly stable ideals are a key tool in commutative algebra and algebraic geometry. These ideals have nice combinatorial properties that makes them well suited for both theoretical and computational applications. In the case of polynomial rings with coefficients in a field with characteristic zero, the notion of strongly stable ideals coincides with the notion of Borel-fixed ideals. Ideals of ...