Parametric Factorization of Matrices
Gaofeng Huang, Frank Kutzschebauch
TL;DR
This survey consolidates parametric factorization results for matrices in $SL_n$ and related classical groups across algebraic, continuous, and holomorphic parameter dependences, using the Oka principle and K-theory as unifying tools. It organizes known factorization results by properties of the coefficient ring $R$ (Euclidean, Bass stable rank 1, Banach algebras) and by dependence type (algebraic, continuous, holomorphic), and extends the discussion to polynomial rings, Stein spaces, and vector-bundle automorphisms. Key contributions include clarified factorization lengths (e.g., four factors in many cases), nullhomotopy characterizations, and the development of universal bounds $L(n,d)$, $K(n,d)$, $K_{sp}(n,d)$ for holomorphic and continuous maps, plus foundational results for algebraic maps via $\mathbb{A}^1$-nullhomotopy. The paper also highlights important counterexamples (e.g., Cohn) and concrete future directions, such as extending results to more groups and refining factor counts.
Abstract
In this survey paper we study parametric versions of writing a matrix in $SL_n (\mathbb{C})$ as a product of lower and upper unitriangular matrices in interchanging order as well as generalizations to other classical groups. We give an account of algebraic, continuous and holomorphic factorization results, from the standpoint of Several Complex Variables. Out of the wealth of algebraic results, we only concentrate on those which are related to holomorphic factorization and often formulate them in a specific form, e.g. for the field of complex numbers in place of more general fields or principal ideal domains. The number of unitriangular matrices needed is a difficult problem and is solved in very specific cases only. We give a new lower bound for factorizing matrices in $SL_2 (\mathbb{C})$ continuously parametrized by two dimensional normal topological spaces.