On the ascending chain condition for mixed difference ideals

TL;DR

This paper proves that the ring of difference polynomials over a difference field does not satisfy the ascending chain condition for mixed difference ideals, resolving an open problem in difference algebra.

Contribution

It demonstrates a fundamental property of difference polynomial rings, showing they lack the ascending chain condition for mixed difference ideals, impacting difference algebraic geometry.

Findings

Ring of difference polynomials does not satisfy the ascending chain condition.

Solves an open problem posed by E. Hrushovski.

Advances understanding of the structure of difference algebraic objects.

Abstract

We show that the ring of difference polynomials over a difference field does not satisfy the ascending chain condition for mixed difference ideals. This result solves an open problem of difference algebra stated by E. Hrushovski in connection with the development of difference algebraic geometry.