Leonid N. Vaserstein

Bass's first stable range condition

Let R be an associative ring with unit. Bass’s lowest (the first) stable range condition [l] asserts the following: if a and b in R satisfy Ra + R& = R, then there exists C in R with a + tb left invertible (that is, R(a + tb) = R; Theorem 2.6 below says that the requirement that a + tb be a unit is not really stronger). More exactly, this is the ‘left’ version of the condition, and there is a symmetric ‘right’ version; but the two versions are in fact equivalent (see Theorem 2.1 below). For brevity, we call a ring satisfying the condition a B-ring. The expression ‘stable range (or rank) of R is l’, or ‘w(R) s I’, or ‘a ring of stable range 1’ is used for this in [19] and other places. In this note known results are collected and a little information is added on Brings. We do not introduce here Bass’s higher stable range conditions (see [I, 3, 9, 11, 13, 14, 16-20, 221) which follow from the first one by [19, Theorem 11, or nonassociative B-rings (see [4, 51, where ‘ring of stable range 2’ means a ring satisfying the first stable range condition). A nice geometric characterization of B-rings is given in [21]. The note was inspired by (and called after) an old unpublished paper by I. Kaplansky which contains Theorems 2.4, 2.6, 2.8, and 5.3, so I dedicate it to him. I thank him for permition to publish his results and pointing out the paper [6] (which contains Theorems 2.2, 2.7, and the equivalence of 5.3(a) with 5.3(d); a ring has ‘substitution property’ in the sense of [6] if and only if it is a B-ring), R. Herman for discussions of the stable range of P-algebras, and a referee for suggestions.

On systems of particles with finite-range and/or repulsive interactions

In an arbitrary system of particles with central repulsive interactions, right and left velocities exist at each moment of time, including infinity. An arbitrary system of particles with finite-range interactions splits into independent bounded clusters. The number of collisions in Sinais billiard is finite.

On a question of M. Newman on the number of commutators

Abstract M. Newman asked whether there is an absolute constant c such that every matrix in SLnR is the product of at most c commutators, where R ranges over euclidean commutative rings and n ⩾ 3. We give here a negative answer. However, if for the ring R every matrix in SLmR is the product of a bounded number of commutators for some fixed m ⩾ 3, then for all sufficiently large n, every matrix in SLnR is the product of six commutators.

Isometric embed-dings between classical Banach spaces, cubature formulas, and spherical designs

AbstractIf an isometric embeddinglpm →lqn with finitep, q>1 exists, thenp=2 andq is an even integer. Under these conditions such an embedding exists if and only ifn⩾N(m, q) where

Commutators and Companion Matrices over Rings of Stable Rank 1

\left( {\begin{array}{*{20}c} {m + q/2 - 1} \\ {m - 1} \\ \end{array} } \right) \leqslant N(m,q) \leqslant \left( {\begin{array}{*{20}c} {m + q - 1} \\ {m - 1} \\ \end{array} } \right).

On the groups SL2(ℤ[x]) and SL2(k[x, y])

To construct some concrete embeddings, one can use orbits of orthogonal representations of finite groups. This yields:N(2,q)=q/2+1 (by regular (q+2)-gon),N(3, 4)=6 (by icosahedron),N(3, 6)⩾11 (by octahedron), etc. Another approach is based on relations between embeddings, Euclidean or spherical designs and cubature formulas. This allows us to sharpen the above lower bound forN(m, q) and obtain a series of concrete values, e.g.N(3, 8)=16 andN(7, 4)=28. In the cases (m, n, q)=(3, 6, 10) and (3, 8, 15) some ε-embeddings with ε ∼ 0.03 are constructed by the orbit method. The rigidness of spherical designs in Bannais sense and a similar property for the embeddings are considered, and a conjecture of [7] is proved for any fixed (m, n, q).

The subnormal structure of general linear groups

For a class of associative rings R with 1 containing every ring which is finitely generated as a module over its center, we obtain a complete description of all subgroups of pseudo-orthogonal groups O2nR which are normalized by elementary orthogonal matrices.

Waring's problem for algebras over fields

Abstract We consider the group GL n A of all invertible n by n matrices over a ring A satisfying the first Bass stable range condition. We prove that every matrix is similar to the product of a lower and upper triangular matrix, and that it is also the product of two matrices each similar to a companion matrix. We use this to show that, when n ⩾3 and A is commutative, every matrix in SL n A is the product of two commutators.