Gauss-Bruhat decomposition as an example of Thomas decomposition

摘要

Both the Gauss-Bruhat decomposition and the LU-decomposition of the general linear group over a field are examples of a Thomas decomposition of systems of polynomial equations and inequations into disjoint triangular systems, a recently rediscovered method of the nineteen-thirties, applied to the inequation det (A) ≠ 0 for an n × n-matrix of indeterminants. More specifically it is shown that the cells of the two decompositions can be described by determinantal equations and inequations yielding simple systems in the sense of Thomas of a rather special type, which are called split and allow counting solutions over any finite field.