Related papers: The Eisenbud-Green-Harris Conjecture for Defect Tw…

The Eisenbud-Green-Harris (EGH) conjecture offers a generalization of the famous Macaulay's theorem about the Hilbert functions of homogeneous ideals in a polynomial ring $K[x_1,\ldots, x_n]$. In this survey paper, we provide a good…

The Eisenbud-Green-Harris conjecture states that a homogeneous ideal in k[x_1,...,x_n] containing a homogeneous regular sequence f_1,...,f_n with deg(f_i)=a_i has the same Hilbert function as an ideal containing x_i^{a_i} for 1 \leq i \leq…

It has been conjectured by Eisenbud, Green and Harris that if $I$ is a homogeneous ideal in $k[x_1,...,x_n]$ containing a regular sequence $f_1,...,f_n$ of degrees $\deg(f_i)=a_i$, where $2\leq a_1\leq ... \leq a_n$, then there is a…

Let $S$ be a standard graded polynomial ring over a field, and $I$ be a homogeneous ideal that contains a regular sequence of degrees $d_1,\ldots,d_n$. We prove the Eisenbud-Green-Harris conjecture when the forms of the regular sequence…

Let $A$ be a graded complete intersection over a field and $B$ the monomial complete intersection with the generators of the same degrees as $A$. The EGH conjecture says that if $I$ is a graded ideal in $A$, then there should be an ideal…

We focus on the structure of a homogeneous Gorenstein ideal $I$ of codimension three in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that $I$ is generated in a fixed degree $d$. For such an ideal $I$ this…

In this paper, we apply liaison theory to the Eisenbud-Green-Harris conjecture and prove that the conjecture holds for a certain subclass of homogeneous ideals in the linkage class of a complete intersection ideal. In the case of three…

Let $K$ be an infinite field and let $I = (f_1,\cdots,f_r)$ be an ideal in the polynomial ring $R = K[x_1,\cdots,x_n]$ generated by generic forms of degrees $d_1,\cdots,d_r$. A longstanding conjecture by Fr\"{o}berg predicts the shape of…

There is a longstanding conjecture by Fr\"oberg about the Hilbert series of the ring $R/I$, where $R$ is a polynomial ring, and $I$ an ideal generated by generic forms. We prove this conjecture true in the case when $I$ is generated by a…

Let $S$ be a polynomial ring over a field and $I\subseteq S$ a homogeneous ideal containing a regular sequence of forms of degrees $d_1, \ldots, d_c$. In this paper we prove the Lex-plus-powers Conjecture when the field has characteristic 0…

We prove an inequality between Hilbert functions of local cohomology modules supported in the homogeneous maximal ideal of standard graded algebras over a field, within the framework of embeddings of posets of Hilbert functions. As a main…

Let $R^h$ denote the polynomial ring in variables $x_1,\,\ldots,\, x_h$ over a specified field $K$. We consider all of these rings simultaneously, and in each use lexicographic (lex) monomial order with $x_1 > \cdots > x_h$. Given a fixed…

Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an…

Macaulay's theorem and Fr\"oberg's conjecture deal with the Hilbert function of homogeneous ideals in polynomial rings $S$ over a field $K$. In this short note we present some questions related to variants of Macaulay's theorem and…

Let $G$ be a finite simple graph on the vertex set $[n] = \{ 1, \ldots, n \}$ and $K[X, Y] = K[x_1, \ldots, x_n, y_1, \ldots, y_n]$ the polynomial ring in $2n$ variables over a field $K$ with each $\mathrm{deg} x_i = \mathrm{deg} y_j = 1$.…

Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$ and $I\subset S$ be a squarefree monomial ideal generated in degree $n-2$. Motivated by the remarkable behavior of the powers of $I$ when $I$ admits a linear resolution, as…

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an N-graded ring generated by…

Let $M$ be an ideal in $K[x_1,...,x_n]$ ($K$ is a field) generated by products of linear forms and containing a homogeneous regular sequence of some length. We prove that ideals containing $M$ satisfy the Eisenbud-Green-Harris conjecture…

Let $d_1,...,d_r$ be positive integers and let $I = (F_1,...,F_r)$ be an ideal generated by general forms of degrees $d_1,...,d_r$, respectively, in a polynomial ring $R$ with $n$ variables. When all the degrees are the same we give a…

The $n$-type vectors introduced by Geramita, Harima and Shin are in 1-1 correspondence with the Hilbert functions Artinian of lex ideals. Letting $\mathbb{A} =\{a_1,..., a_n\}$ define the degrees of a regular sequence, we construct ${\rm…