Related papers: Holes in I^n

This paper introduces a novel approach to the axiomatic theory of quadratic forms. We work internally in a category of certain partially ordered sets, subject to additional conditions which amount to a strong form of local presentability.…

Let $G$ be a group which is the semidirect product of a normal subgroup $N$ and some subgroup $T$. Let $I^n(G)$, $n\ge 1$, denote the powers of the augmentation ideal $I(G)$ of the group ring $\Z(G)$. Using homological methods the groups…

Let $n$ be any positive integer and $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ be the semigroup of all order isomorphisms between principal filters of the $n$-th power of the set of positive integers $\mathbb{N}$ with the product order. We study…

The set of all maximal ideals of the ring $\mathcal{M}(X,\mathcal{A})$ of real valued measurable functions on a measurable space $(X,\mathcal{A})$ equipped with the hull-kernel topology is shown to be homeomorphic to the set $\hat{X}$ of…

Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive…

Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$. Let $I$ be a homogeneous ideal of $A$ with $I \ne A$ and $H_{A/I}$ the Hilbert function of the quotient algebra $A / I$. Given…

Let $R=\mathbb{F}_p[x_1,\ldots,x_n]$ and let $\mathbf{F}$ be the ring of Frobenius operators over $R$. We introduce a notion of Bernstein dimension and multiplicity for the class of finitely generated $\mathbf{F}$-modules whose structure…

Let $\Gamma$ be the Fuchsian group of the first kind. For an even integer $m\ge 4$, we study $m/2$-holomorphic differentials in terms of space of (holomorphic) cuspidal modular forms $S_m(\Gamma)$. We also give in depth study of Wronskians…

Let $W$ be a finite Coxeter group. We give an algebraic presentation of what we refer to as ``the non-crossing algebra'', which is associated to the hyperplane complement of $W$ and to the cohomology of its Milnor fibre. This is used to…

We show that subsets of $\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in…

Let $R$ and $S$ be polynomial rings of positive dimensions over a field $k$. Let $I\subseteq R, J\subseteq S$ be non-zero homogeneous ideals none of which contains a linear form. Denote by $F$ the fiber product of $I$ and $J$ in…

Let $A$ be a Noetherian ring and let $I$ be an ideal in $A$. Let $\mathcal{F} = \{ J_n \}_{n \geq 0}$ be a multiplicative filtration of ideals in $A$ such that $\mathcal{R}(\mathcal{F}) = \bigoplus_{n \geq 0} J_n$ is a finitely generated…

We give a simple way to study the isotypical components of the homology of simplicial complexes with actions of finite groups, and use it for Milnor fibers of ICIS. We study the homology of images of mappings $f_t$ that arise as…

The $12$-dimensional Fomin-Kirillov algebra $FK_3$ is defined as the quadratic algebra with generators $a$, $b$ and $c$ which satisfy the relations $a^2=b^2=c^2=0$ and $ab+bc+ca=0=ba+cb+ac$. By a result of A. Milinski and H.-J. Schneider,…

The dimension spectrum of a conformal iterated function system (CIFS) is the set of all Hausdorff dimensions of its various subsystem limit sets. This brief note provides two constructions -- (i) a compact perfect set that cannot be…

We show that for any integer n and any field k of characteristic different from 2 there are at most finitely many isomorphism classes of quadratic morphisms from the projective line over k to itself with a finite postcritical orbit of size…

For each positive integer n this paper considers a one-parameter family of complex-valued measures on the symmetric group S_n, depending on a complex parameter z. For parameter values z=p^f a prime power, this measure describes splitting…

Let $F$ be a non-zero polynomial with integer coefficients in $N$ variables of degree $M$. We prove the existence of an integral point of small height at which $F$ does not vanish. Our basic bound depends on $N$ and $M$ only. We separately…

It was conjectured by Furstenberg that for any $x\in [0,1]\backslash Q$, $$ \dim_H \bar{\{2^nx ({\text{mod}}\ 1): n\ge 1\}}+ \dim_H \bar{\{3^nx ({\text{mod}}\ 1): n\ge 1\}}\ge 1. $$ When $x$ is a normal number, the above result holds…

We show that if X_n is a variety of cxn-matrices that is stable under the group Sym([n]) of column permutations and if forgetting the last column maps X_n into X_{n-1}, then the number of Sym([n])-orbits on irreducible components of X_n is…