On the Whitehead determinant for semi-local rings

AbstractWe answer the question: when the Whitehead determinant of a semi-local ring is the abelization of the multiplicative group

Parametrization of Pythagorean triples by a single triple of polynomials

It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials.Comment: to appear in J. Pure Appl. Algebr

On the best choice of a damping sequence in iterative optimization methods

Some iterative methods of mathematical programming use a damping sequence {αt} such that 0 _< αt < 1 for all t, at - 0 as t - ∞, and Σαt = ∞. For example, αt = 1l(t + 1) in Brown's method for solving matrix games. In this paper, for a model class of iterative methods, the convergente rate for any damping sequence {αt}depending only on time t is computed. This computation is used to find the best damping sequence

Prestabilization for K1 of banach algebras

AbstractThe injective stability for the general linear group modulo elementary matrices begins at one plus the stable range of the ring of entries. At one step earlier, the kernel of stabilization is perhaps larger than the group of elementary matrices. Using a Dieudonné-style determinant, it is shown that this kernel is generated by matrices of the form (1 + XY)(1 + XY), under certain conditions on the ring of entries, or in the relative case, on the ideal. For any ideal of stable rank one, the kernel is given in terms of generators (X + Z + XYZ)(X + Z + ZYX). Under somewhat stronger conditions, the kernel is shown to be a commutator subgroup

Normal Subgroups of Classical Groups over von Neumann Regular Rings

AbstractLet R be a von Neumann regular ring. We obtain a complete description of all subgroups of the pseudo-orthogonal groups O2nR for n ≥ 3 which are normalized by elementary orthogonal matrices. We also prove the normality of EO2n(R, J) in O2nR when R is an abelian von Neumann regular ring and n ≥ 1

Solving quadratic equations over polynomial rings of characteristic two

We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain A with 1 and a polynomial equation an tn + ··· + a0 = 0 with coefficients ai in A, our problem is to find its roots in A. We show that when A = B[x] is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over B. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over A when A is F[x1,... ,xN ] or F(x1,... ,xN ) for any finite field F and any number N of variables. The case of quadratic equations in characteristic two is studied in detail

Similarity Classes of 3×33\times 3 Matrices over a Local Principal Ideal Ring

In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.Mathematic

On Quasi-homomorphisms and Commutators in the Special Linear Group over a Euclidean Ring

We prove that for any euclidean ring R and n at least 6, Gamma=SL_n(R) has no unbounded quasi-homomorphisms. From Bavard's duality theorem, this means that the stable commutator length vanishes on Gamma. The result is particularly interesting for R = F[x] for a certain field F (such as the field C of complex numbers, because in this case the commutator length on Gamma is known to be unbounded. This answers a question of M. Ab\'ert and N. Monod for n at least 6.Comment: This is the final version. 8 pages; title changed again; title changed, a little generalization of the main theore