Un théorème de factorisation des produits d'endomorphismes de Kn

Abstract

Let H be a finite set of endomorphisms of a finite dimension vector space K n over a field K. We prove that if φ is a product φ 1 φ 2 ··· φ m of endomorphisms of K n belonging to H , we can extract of it a product φ i φ i + 1 ··· φ j = ψ which is a pseudoregular endomorphism. That is: ψ has a representation as a square matrix over K which is diagonal sum of a null matrix and of a regular one. This result holds also if K is a skew-field. As a consequence of this result, we obtain two theorems about the rational power series in noncommuting variables. The first concerns the “regularity” of the support of the rational power series. The second generalizes a theorem of Polyà [12] and another one of Cantor [2], about rational power series with algebraic integer coefficients.