Related papers: Binomial expansion and the $\mathrm{v}$-number

Let $K$ be a field, $I\subset R=K[x_1,\dots,x_n]$ and $J\subset T=K[y_1,\dots,y_m]$ be graded ideals. Set $S=R\otimes_KT$ and let $L=IS+JS$. The behaviour of the $\text{v}$-function $\text{v}(L^k)$ in terms of the $\text{v}$-functions…

Let $k$ be a field, let $A$ and $B$ be polynomial rings over $k$, and let $S= A \otimes_k B$. Let $I \subseteq A$ and $J \subseteq B$ be monomial ideals. We establish a binomial expansion for rational powers of $I+J \subseteq S$ in terms of…

In this article, we investigate the $\operatorname{v}$-numbers of powers of monomial ideals and their integral closures in a polynomial ring $S$. We provide an alternative proof for determining the $\operatorname{v}$-numbers of powers of…

Let $I$ be a graded ideal of a standard graded polynomial ring $S$ with coefficients in a field $K$. The asymptotic behaviour of the $\text{v}$-number of the powers of $I$ is investigated. Natural lower and upper bounds which are linear…

We introduce an exact functor defined on multigraded modules which we call the expansion functor and study its homological properties. The expansion functor applied to a monomial ideal amounts to substitute the variables by monomial prime…

Given arbitrary monomial ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $K$, we investigate the Stanley depth of powers of the sum $I+J$, and their quotient rings, in $A\otimes_K B$ in terms of those of $I$ and $J$. Our…

In this paper, we give formulas for $v$-number of edge ideals of some graphs like path, cycle, 1-clique sum of a path and a cycle, 1-clique sum of two cycles and join of two graphs. For an $\mathfrak{m}$-primary monomial ideal $I\subset…

Let $A, B$ be polynomial rings over a field $k$, and $I\subseteq A, J\subseteq B$ proper homogeneous ideals. We analyze the associated primes of powers of $I+J\subseteq A\otimes_k B$ given the data on the summands. The associated primes of…

This paper investigates the v-number of various classes of monomial ideals. First, we considers the relationship between the v-number and the regularity of the mixed product ideal $I$, proving that $\mathrm{v}(I) \leq \mathrm{reg}(S/I)$.…

We introduce and study rational symbolic powers of ideals in Noetherian rings. We give membership criteria for rational symbolic powers and discuss settings where they agree with integer symbolic powers. We investigate the binomial…

Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms…

Let $I$ and $J$ be nonzero ideals in two Noetherian algebras $A$ and $B$ over a field $k$. Let $I+J$ denote the ideal generated by $I$ and $J$ in $A\otimes_k B$. We prove the following expansion for the symbolic powers: $$(I+J)^{(n)} =…

In this paper we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness and having linear quotients are preserved under taking…

M. Morales introduced a family of binomial ideals that are binomial extensions of square free monomial ideals. Let $I\subset \si$ be a square free monomial ideal and $J\subset\sis$ a sum of scroll ideals with some extra conditions, we…

There are two different notions for symbolic powers of ideals existing in the literature, one defined in terms of associated primes, the other in terms of minimal primes. Elaborating on an idea known to Eisenbud, Herzog, Hibi, and Trung, we…

Let $I\subset S$ be a graded ideal of a standard graded polynomial ring $S$ with coefficients in a field $K$, and let $\text{v}(I)$ be the $\text{v}$-number of $I$. In previous work, we showed that for any graded ideal $I\subset S$…

We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and…

Given arbitrary homogeneous ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $k$, we investigate the depth and the Castelnuovo-Mumford regularity of powers of the sum $I+J$ in $A \otimes_k B$ in terms of those of $I$ and $J$.…

We generalize some results of $\mathrm{v}$-number for arbitrary monomial ideals by showing that the $\mathrm{v}$-number of an arbitrary monomial ideal is the same as the $\mathrm{v}$-number of its polarization. We prove that the…

In this paper, we derive formal general formulas for noncommutative exponentiation and the exponential function, while also revisiting an unrecognized, and yet powerful theorem. These tools are subsequently applied to derive counterparts…