Related papers: A Survey on The Eisenbud-Green-Harris Conjecture

We introduce and study equivariant Hilbert series of ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid $Inc(\mathbb{N})$ of strictly increasing functions.…

In this note we show that Harbourne's conjecture is true for symbolic powers of ideals of points, we check that the stable version of this conjecture is valid for ideals of very general points (resp. generic points) in $\mathbb P_{\mathbb…

One studies the structure of the Rees algebra of an almost complete intersection monomial ideal of finite co-length in a polynomial ring over a field, assuming that the least pure powers of the variables contained in the ideal have the same…

A simply connected topological space is called \emph{rationally elliptic} if the rank of its total homotopy group and its total (co)homology group are both finite. A well-known Hilali conjecture claims that for a rationally elliptic space…

The equivariant Hilbert series of an ideal generated by an orbit of a monomial under the action of the monoid $\mbox{Inc}(\mathbb{N})$ of strictly increasing functions is determined. This is used to find the dimension and degree of such an…

We formulate and prove a version of the Segal Conjecture for infinite groups. For finite groups it reduces to the original version. The condition that G is finite is replaced in our setting by the assumption that there exists a finite model…

Let $R$ be the face ring of a simplicial complex of dimension $d-1$ and ${\mathcal R}(\mathfrak{n})$ be the Rees algebra of the maximal homogeneous ideal $\mathfrak{n}$ of $R.$ We show that the generalized Hilbert-Kunz function…

This paper presents some considerations about the Goldbach's conjecture (GC). The work is based on elementary results of the number theory and it provides a constructive method that permits, given an even integer, to find at least a pair of…

We prove the Farrell-Jones Isomorphism Conjecture about the algebraic K-theory of a group ring RG in the case where the group G is the fundamental group of a closed Riemannian manifold with strictly negative sectional curvature. The…

In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in…

The holomorphy conjecture predicts that the local Igusa zeta function associated to a hypersurface and a character is holomorphic on $\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local…

This paper gives new bounds on the first Hilbert coefficient of an ideal of finite colength in a Cohen-Macaulay local ring. The bound given is quadratic in the multiplicity of the ideal. We compare our bound to previously known bounds, and…

We give conjectures on the "asymptotic" behaviour of the Hilbert series of (quotients by) generic ideals in the exterior algebra, as the number of variables tend to infinity. Our conjectures are supported by extensive computer calculations.

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal $I$ in the polynomial ring $S=K[x_1,...,x_n]$ and a finitely generated graded $S$-module, the Hilbert coefficients $e_i(M/I^kM)$ are polynomial…

A homotopy theoretic description is given for trivial unit conjecture in the group ring ZG.

Let G be a reductive algebraic group and H a closed subgroup of G. Explicit constructions of G-invariant ideals in the algebra K[G/H] are given. This allows to obtain an elementary proof of Matsushima's criterion: a homogeneous space G/H is…

In this paper, we prove an averaged version of an algebraicity conjecture in \cite{GKZ87} concerning the values of higher Green's function at CM points. Furthermore, we give the factorization of the ideal generated by such algebraic value…

Let $(R, \mathfrak{m}, \Bbbk)$ be a Noetherian three-dimensional Cohen-Macaulay analytically unramified ring and $I$ an $\mathfrak{m}$-primary $R$-ideal. Write $X = \mathrm{Proj}\left(\oplus_{n \in \mathbb{N}} \overline{I^n}t^n\right)$. We…

We introduce a new class of polynomial ideals associated to a simple graph, $G$. Let $K[E_G]$ be the polynomial ring on the edges of $G$ and $K[V_G]$ the polynomial ring on the vertices of $G$. We associate to $G$ an ideal, $I(X_G)$,…

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial…