The ascending chain condition for radical well-mixed difference ideals

Abstract

We establish a finite-basis theorem for radical well-mixed difference ideals in a difference polynomial ring with one endomorphism over an arbitrary difference field. The result extends to every finitely difference-generated difference algebra over such a field and is equivalent to the ascending chain condition for radical well-mixed difference ideals, hence resolving a long-standing open problem in difference algebra. The proof proceeds in two stages. Over an inversive difference coefficient field, we establish a uniform inverse-transform bound relating the perfect closure of a finite system to its radical well-mixed closure. This uses Noetherian induction in a fixed truncated polynomial ring, together with generic freeness and going-down for flat ring homomorphisms. We then prove that the radical well-mixed closure of a system over a difference field $K$ is the contraction of its radical well-mixed closure over the inversive closure of $K$. This contraction result follows from successive prime lifting along flat extensions and removes the inversiveness assumption without requiring the coefficient-field extension to be finitely generated.