Related papers: Forcing the weak Lefschetz property for equigenera…
In [MMR], Micha\l{}ek--Mir\'o-Roig give a beautiful geometric characterization of Artinian quotients by ideals generated by quadratic or cubic monomials, such that the multiplication map by a general linear form fails to be injective in the…
We construct new families of Artinian Gorenstein graded $K$-algebras of arbitrary codimension having binomial Macaulay dual generators and satisfying the weak or the strong Lefschetz property. This is a companion paper to \cite{ADFMMSV},…
To any lattice $L \subset \mathbb{Z}^{m}$ one can associate the lattice ideal $I_{L} \subset K[x_{1},...,x_{m}]$. This paper concerns the study of the relation between the binomial arithmetical rank and the minimal number of generators of…
Our starting point is a basic problem in Hermite interpolation theory, namely determining the least degree of a homogeneous polynomial that vanishes to some specified order at every point of a given finite set. We solve this problem if the…
Every artinian quotient of $K[x,y]$ has the strong Lefschetz property if $K$ is a field of characteristic zero or is an infinite field whose characteristic is greater than the regularity of the quotient. We improve this bound in the case of…
Let $A=\{{\bf a}_1,...,{\bf a}_m\} \subset \mathbb{Z}^n$ be a vector configuration and $I_A \subset K[x_1,...,x_m]$ its corresponding toric ideal. The paper consists of two parts. In the first part we completely determine the number of…
In this paper we study the Freiman inequality for the minimal number of generators of the square of an equigenerated monomial ideal. Such an ideal is called a Freiman ideal if equality holds in the Freiman inequality. We classify all…
An equigenerated monomial ideal $I$ in the polynomial ring $S= K[x_1,\ldots,x_n]$ is a Freiman ideal if $\mu(I^2)=\ell(I)\mu(I)-{\ell(I)\choose 2}$ where $\ell(I)$ is the analytic spread of $I$ and $\mu(I)$ is the number of minimal…
Given any equigenerated monomial ideal $I$ with the property that the defining ideal $J$ of the fiber cone $ F(I)$ of $I$ is generated by quadratic binomials, we introduce a matrix such that the set of its binomial $2$-minors is a…
Let $I$ be a homogeneous Artinian ideal in a polynomial ring $R=k[x_1,...,x_n]$ over a field $k$ of characteristic 0. We study an equivalent condition for the generic initial ideal $\gin(I)$ with respect to reverse lexicographic order to be…
We obtain tight bounds for the minimal number of generators of an ideal with bounded-degree generators in a polynomial ring $K[X_1,\dots,X_n],$ as well as a sharp quantification of the maximum possible size of a minimal generating set of…
Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the…
In a recent paper, Schenck and Seceleanu showed that in three variables, any ideal generated by powers of linear forms has the Weak Lefschetz Property (WLP). This result contrasts with examples, in our previous work, of ideals in four…
An equigenerated monomial ideal $I$ is a Freiman ideal if $\mu(I^2)=\ell(I)\mu(I)-{\ell(I)\choose 2}$ where $\ell(I)$ is the analytic spread of $I$ and $\mu(I)$ is the least number of monomial generators of $I$. Freiman ideals are special…
Let K be an algebraically closed field of characteristic zero and let I=(f_1,...,f_n) be a homogeneous R_+-primary ideal in R:=K[X,Y,Z]. If the corresponding syzygy bundle Syz(f_1,...,f_n) on the projective plane is semistable, we show that…
This paper initiates a systematic study for key properties of Artinian Gorenstein \(K\)-algebras having binomial Macaulay dual generators. In codimension 3, we demonstrate that all such algebras satisfy the strong Lefschetz property, can be…
Stanley proved that, in characteristic zero, all artinian monomial complete intersections have the strong Lefschetz property. We provide a positive characteristic complement to Stanley's result in the case of artinian monomial complete…
Recently, H. Dao and R. Nair gave a combinatorial description of simplicial complexes $\Delta$ such that the squarefree reduction of the Stanley-Reisner ideal of $\Delta$ has the WLP in degree $1$ and characteristic zero. In this paper, we…
We reprove and generalize a result of Moh which gives a lower bound on the minimal number of generators of an ideal in a power series ring in three variables x,y,z over a field k. As a consequence, in each characteristic of the field k, we…
Two minimal generating sets of the first syzygies of a monomial ideal are produced, given the minimal generating set of the ideal.