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.
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BibTeX
@misc{li2026ascending,
title = {{The ascending chain condition for radical well-mixed difference ideals}},
author = {Wei Li and Jie Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34389},
eprint = {2610.34389},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34389},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The ascending chain condition for radical well-mixed difference ideals
AU - Wei Li
AU - Jie Wang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34389
UR - https://arxiv.org/abs/2610.34389
AB - 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.
ER -