Reduction of a matrix depending on parameters to a diagonal form by addition operations

Abstract

ABSTRACT. It is shown that any n by n matrix with determinant 1 whose entries are real or complex continuous functions on a finite dimensional normal topological space can be reduced to a diagonal form by addition operations if and only if the corresponding homotopy class is trivial, provided that n j¿2 for real-valued functions; moreover, if this is the case, the number of operations can be bounded by a constant depending only on n and the dimension of the space. For real functions and n = 2, we describe all spaces such that every invertible matrix with trivial homotopy class can be reduced to a diagonal form by addition operations as well as all spaces such that the number of operations is bounded.

Parametric jet interpolation for Stein manifolds with the density property

“…The existence of a continuous lift was proven by Vaserstein [Vas88]. To find a holomorphic lift, the authors use the Oka-Grauert-Gromov principle for sections of holomorphic submersions coming from the diagram…”

Factorization of symplectic matrices into elementary factors

“…The second author and E.Doubtsov recently proved that the result holds for rings with Bass stable rank 1 ( [DoKu]) If R is a unital commutative Banach algebra then every null-homotopic matrix in SL n (R) is in E n (R) ( [Mil71]). In the case of R = C(X), the continuous complex functions on a finite dimensional normal topological space, Vaserstein had previously proven the same result for null-homotopic matrices ( [Vas88]). Finally, the first two authors ([IK12]) proved the result for null-homotopic matrices in the case of R = O(X), the holomorphic functions on a reduced Stein space X, thus solving the so-called Vaserstein problem of Gromov ([Gro89]).…”

“…For n ≥ 2 Kopeiko proved this for R = k[x 1 , · · · , x k ] ([Kop78]) and Grunewald/Mennicke/ Vaserstein proved it for R = Z[x 1 , · · · , x k ]. In this paper we take up the study for various function spaces and we prove symplectic versions of the results in [Mil71], [DoKu] and [Vas88]. The Vaserstein problem for null-homotopic holomorphic symplectic matrices turns out to be very complicated and requires the use of Gromov's Oka principle for holomorphic sections of elliptic bundles ([Gro89]).…”

On the Bass Stable Rank of Stein Algebras

“…Further, it was proved by Vaserstein in [V3,Th. 4] that for A = C(X), the algebra of complex-valued continuous functions on a d-dimensional normal topological space X, E n (A) coincides with the set of null-homotopic maps, i.e., maps in SL n (A) := C(X, SL n (C)) homotopic (in this class of maps) to a constant map, and, moreover, there exists a constant v(d) ∈ N depending on d only such that sup n t n (A) ≤ v(d) (see also [DV,Lm.…”

“…A similar problem for the algebra O(X), where X is a finite-dimensional reduced Stein space, was posed by Gromov in [G, 3.5.G] (the paper is devoted to the extension of the classical Oka-Grauert theorem) and solved recently by Ivarsson and Kutzschebauch in [IK1] (see also [IK2]) based on [V3,Th. 4] and [F,Th.…”