On the Hilbert series of ideals generated by generic forms

There is a longstanding conjecture by Fröberg 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 large number of forms, all of the same degree. We also conjecture that an ideal generated by m’th powers of generic forms of degree d≥2 gives the same Hilbert series as an ideal generated by generic forms of degree md. We verify this in several cases. This also gives a proof of the first conjecture in some new cases.     

On ideals generated by two generic quadratic forms in the exterior algebra

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 upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms.     

Density versions of Schur's theorem for ideals generated by submeasures

We characterize ideals of subsets of natural numbers for which some versions of Schur's theorem hold. These are similar to generalizations shown by Bergelson (1986) in [1] and Frankl, Graham and Rödl (1990) in [7]. Additionally, we prove a generalization of an iterated version of Ramsey's theorem.     

Finite basis for radical well-mixed difference ideals generated by binomials

In this paper, we prove a finite basis theorem for radical well-mixed difference ideals generated by binomials. As a consequence, every strictly ascending chain of radical well-mixed difference ideals generated by binomials in a difference polynomial ring is finite, which solves an open problem in difference algebra raised by Hrushovski in the binomial case.     

A criterion for the primeness of ideals generated by polynomials with separated variables

Proving primeness of an idealI=〈f ₁, …,f _m〉 in a polynomial ringR=K[X ₁, …,X _n]ofn indeterminates over an algebraically closed fieldK is a difficult task in general. Although there are straightforward algorithms that decide whetherI is prime or not, they are prohibitively lengthy if the number of indeterminates or the degrees of thef _iare large. In this paper we will give an easy criterion for the primeness ofI if thef _iare polynomials with separated variables, i.e. no mixed monomials occur in thef _i. The work on this paper was done while the author was a MINERVA fellow at Tel Aviv University.     

Right ideals generated by an idempotent of finite rank

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let f(X₁,…,X_t) be an arbitrary and fixed polynomial over K in noncommuting indeterminates X₁,…,X_t with constant term 0 such that for some μK occurring in the coefficients of f(X₁,…,X_t). It is proved that a right ideal ρ of R is generated by an idempotent of finite rank if and only if the rank of f(x₁,…,x_t) is bounded above by a same natural number for all x₁,…,x_tρ. In this case, the rank of the idempotent that generates ρ is also explicitly given. The results are then applied to considering the triangularization of ρ and the irreducibility of f(ρ), where f(ρ) denotes the additive subgroup of R generated by the elements f(x₁,…,x_t) for x₁,…,x_tρ.     

Projective generation of ideals in a certain ring

Let R be an affine domain of dimension $n\ge 3$ over a field of characteristic 0 and $D=R[X,Y]/(XY)$. Let $I?D$ be a local complete intersection ideal of height n such that $\mu (I/I^2)=n$. This paper examines under what condition I is surjective image of a projective D-module of rank n.     

An invariant cubature formula of degree nine for a hypercube

In this work, cubature formulas for computation of integrals over the hypercube in R ⁿ

$C_n = \{ x = (x_1 ,x_2 ,...,x_n ) \in R^n | - 1 \leqslant x_i \leqslant 1,i = 1,2,...,n\} $

are constructed using Sobolev?s theorem. These formulas are precise for all polynomials of degree at most nine and are invariant under the group of all orthogonal transformations of the hyperoctahedron

$G_n = \left\{ {x = (x_1 ,x_2 ,...,x_n ) \in R^n |\sum\limits_{i = 1}^n {|x_i |} \leqslant 1} \right\}$

onto itself.

Section 1 contains introduction into the subject and a review of known results. In Sections 2 and 3, we determine parameters of the cubature formula for n = 4 and n = 3, respectively. Numerical results (the nodes and coefficients of the cubature formulas) are presented in Section 4.     

Spaces of modular forms generated by eta-quotients

In this paper, we study the spaces of modular forms on Γ₀(N) generated by eta-quotients, where the genus of Γ₀(N) is zero or N is a prime. Our results give an answer to a question of Ono (2004, Problem 1.68). 2000 Mathematics Subject Classification Primary—11F20, 11F11     

On the multiplicity of blow-up rings of ideals generated by d-sequences

We consider the Rees ring, the extended Rees ring and the associated graded ring of an ideal which is generated by a homogenous d-sequence. We introduce on these rings certain fine filtrations which allow us to compute their multiplicity with respect to their unique graded maximal ideal.     

Priifer domains with class group generated by the classes of the invertible maximal ideals

In [2] and [3] we described the double coset decomposition for the modular group SL_r+s (Z) with respect to the congruence subgroup Р _r,s (m) for any positive integer m. In the present paper we obtain the double coset decomposition of SL_r+s (R) with respect to Р _r,s (I). Where R is a commutative ring with unioty 1, and I is a nonzero ideal in R such that R/I is the direct sum of local principal ideal rings; thus generalize the results obtained in [2] and [3].