Related papers: Note on bounds for multiplicities

Suppose $A=k[X_1, X_2, \ldots, X_n]$ is a polynomial ring over a field $k$ and $I$ is an ideal in $A$. Then M. P. Murthy conjectured that $\mu(I)=\mu(I/I^2)$, where $\mu$ denotes the minimal number of generators. Recently, Fasel \cite{F}…

The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at…

We extend a result by Huneke and Watanabe bounding the multiplicity of $F$-pure local rings of prime characteristic in terms of their dimension and embedding dimensions to the case of $F$-injective, generalized Cohen-Macaulay rings. We then…

Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\mathrm{deg}\ x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series of $S/I$ is of the form…

Let K be a number field and let S be a finite set of places of K which contains all the Archimedean places. For any f(z) in K(z) of degree d at least 2 which is not a d-th power in \bar{K}(z), Siegel's theorem implies that the image set…

In 2016 Ananyan and Hochster proved Stillman's conjecture by showing the existence of a uniform upper bound on the length of an $R_\eta$-sequence containing fixed $n$ forms of degree at most $d$ in polynomial rings over a field. This result…

A 1993 result of Alon and F\"uredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to…

We show that the graded maximal ideal of a graded $K$-algebra $R$ has linear quotients for a suitable choice and order of its generators if the defining ideal of $R$ has a quadratic Gr\"obner basis with respect to the reverse lexicographic…

We generalize a result of Ein-Lazarsfeld-Smith (math.AG/0202303), proving that for an arbitrary sequence of zero-dimensional ideals, the multiplicity of the sequence is equal with its volume. This is done using a deformation to monomial…

Let G be a connected linear algebraic group over an algebraically closed field k, and let H be a connected closed subgroup of G. We prove that the homogeneous variety G/H is a rational variety over k whenever H is solvable, or when dim(G/H)…

Let (R,m) be an n-dimensional regular local ring, essentially of finite type over a field of characteristic zero. In this paper we study the relationship between the singularities of the scheme defined by an m-primary ideal I of R and the…

In this paper we show some multiplicity estimates theorems for a connected algebraic group (not necessarily commutative) $G$ over an algebraically closed subfield of $\mathbb{C}$. More specifically, under particular assumptions on the…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We study the relations of the index of reducibility and the irreducible multiplicity of an $\mathfrak{m}$-primary ideal of $R$ and these of…

We introduce a spectrum for arbitrary varieties. This generalizes the definition by Steenbrink for hypersurfaces. In the isolated complete intersection singularity case, it coincides with the one given by Ebeling and Steenbrink except for…

For a given algebraically closed field $k$ of characteristic $p>0$ we consider the set ${\mathcal C}_k$, of graded isomorphism classes of {\em standard graded pairs} $(R, I)$, where $R$ is a standard graded ring over the field and $I$ is a…

It has been proved recently by Moreto and Craven that the order of a finite group is bounded in terms of the largest multiplicity of its irreducible character degrees. A conjugacy class version of this result was proved for solvable groups…

We obtain effective bounds on the heights of algebraic integers whose orbits contain multiplicatively dependent values modulo S-integers. Our method is based on a new upper bound on the so-called S-height of polynomial values over the ring…

When M is a finitely generated graded module over a standard graded algebra S and I is an ideal of S, it is known from work of Cutkosky, Herzog, Kodiyalam, R\"omer, Trung and Wang that the Castelnuovo-Mumford regularity of I^mM has the form…

We associate a family of ideal sheaves to any Q-effective divisor on a complex manifold, called higher multiplier ideals, using the theory of mixed Hodge modules and V-filtrations. This family is indexed by two parameters, an integer…

We prove a geometric criterion for the bounded multiplicity property of "small" infinite-dimensional representations of real reductive Lie groupsin both induction and restrictions. Applying the criterion to symmetric pairs, we give a full…