On stabilization and the existence of coprime factorizations

Abstract

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> be an integral domain and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</tex> its quotient field. We show there are plants over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</tex> which have no stable coprime factorizations, but can be stabilized in a stable closed loop. This answers a question posed by Vidyasagar, Schneider, and Francis.