Limits of Cartier divisors

Consider a one-parameter family of algebraic varieties degenerating to a reducible one. Our main result is a formula for the fundamental cycle of the limit subscheme of any family of effective Cartier divisors. The formula expresses this cycle as a sum of Cartier divisors, not necessarily effective, of the components of the limit variety.     

Cartier divisors and geometry of normalG-varieties

We study Cartier divisors on normal varieties with the action of a reductive groupG. We give criteria for a divisor to be Cartier, globally generated and ample, and apply them to a study of the local structure and the intersection theory of aG-variety. In particular, we prove an integral formula for the degree of an ample divisor on a variety of complexity 1, and apply this formula to computing the degree of a closed 3-dimensional orbit in any SL₂-module.Supported by CRDF grant RM1-206 and INTAS grant INTAS-OPEN-97-1570     

Cartier divisors versus invertible sheaves

We consider the original version of the author's example of an algebraic scheme carrying an invertible sheaf arising from no Cartier divisor.     

Remarks on the existence of Cartier divisors

We characterize those invertible sheaves on a noetherian scheme which are definable by Cartier divisors and correct an erroneous counterexample in the literature.     

Radicals in skew polynomial and skew Laurent polynomial rings

We provide a general procedure for characterizing radical-like functions of skew polynomial and skew Laurent polynomial rings under grading hypotheses. In particular, we are able to completely characterize the Wedderburn and Levitzki radicals of skew polynomial and skew Laurent polynomial rings in terms of ideals in the coefficient ring. We also introduce the T-nilpotent radideals, and perform similar characterizations.     

Distributive skew Laurent polynomial rings

We describe skew Laurent polynomial rings that are right distributive.     

Descriptions of radicals of skew polynomial and skew Laurent polynomial rings

In this paper, we first characterize the Levitzki radical of a skew (Laurent) polynomial ring by the prime ideals and skewed prime ideals in the base ring. We next provide formulas for the strongly prime radical and the uniformly strongly prime radical of skew (Laurent) polynomial rings.     

Inverse polynomial expansions of Laurent series

An algorithm is introduced, and shown to lead to various unique series expansions of formal Laurent series, as the sums of reciprocals of polynomials. The degrees of approximation by the rational functions which are the partial sums of these series are investigated. The types of series corresponding to rational functions themselves are also characterized.     

Orthogonality and recurrence for ordered Laurent polynomial sequences

We consider orderings of nested subspaces of the space of Laurent polynomials on the real line, more general than the balanced orderings associated with the ordered bases {1,z⁻¹,z,z⁻²,z²,…} and {1,z,z⁻¹,z²,z⁻²,…}. We show that with such orderings the sequence of orthonormal Laurent polynomials determined by a positive linear functional satisfies a three-term recurrence relation. Reciprocally, we show that with such orderings a sequence of Laurent polynomials which satisfies a recurrence relation of this form is orthonormal with respect to a certain positive functional.     

Growth of generalized Weyl algebras over polynomial algebras and Laurent polynomial algebras

We study the growth and the Gelfand-Kirillov dimension(GK-dimension) of the generalized Weyl algebra(GWA) A = D(σ, a), where D is a polynomial algebra or a Laurent polynomial algebra. Several necessary and sufficient conditions for GKdim(A) = GKdim(D) + 1 are given. In particular, we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates, i.e., GKdim(A) is either 3 or ∞ in this case. Our results generalize several existing results in the literature and ca...     

On common unital divisors of polynomial matrices

We establish a condition for the existence of common unital divisors of polynomial matrices over an arbitrary field, with the divisors having a prescribed characteristic polynomial. The results obtained are applied to find a common solution of matrix polynomial equations.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 20–24.     

A new method for computing polynomial greates common divisors and polynomial remainder sequences

Summary A new method is presented for the computation of a greatest common divisor (gcd) of two polynomials, along with their polynomial remainder sequence (prs). This method is based on our generalization of a theorem by Van Vleck [12] and uniformly treats both normal and abnormal prs's, making use of Bareiss's [3] integer-preserving transformation algorithm for Gaussian elimination. Moreover, for the polynomials of the prs's, this method provides the smallest coefficients that can be expected without coefficient ged computations (as in Bareiss [3]) and it clearly demonstrates the divisibility properties; hence, it combines the best of both the reduced and the subresultant prs algorithms.This paper is affectionately dedicated to the memory of my father     

On extensions of polynomial functions

Let X, Y be two linear spaces over the field ? of rationals and let D ≠ ? be a (?—convex subset of X. We show that every function ?: D → Y satisfying the functional equation $${\mathop\sum^{n+1}\limits_{j=0}}(-1)^{n+1-j}\Bigg(^{n+1}_{j}\Bigg)f\Bigg((1-{j\over {n+1}})x+{j\over{n+1}}y\Bigg)=0,\ \ \ x,y\in\ D,$$ admits an extension to a function F: X → Y of the form $$F(x)=A^o+A^1(x)+\cdot\cdot\cdot+A^n(x),\ \ \ x\in\ X,$$ where A ^o ∈ Y, A^k(x) ? A_k(x,…,x), x ∈ X, and the maps A _k: X ^k → Y are k—additive and symmetric, k ∈ {1,…, n}. Uniqueness of the extension is also discussed.