The polynomial closure is not topological

TL;DR

This paper investigates the polynomial closure of pseudo-convergent sequences in valuation domains, demonstrating that such closures are never topological in domains of rank two or higher, thereby revealing fundamental limitations in their topological structure.

Contribution

The paper characterizes polynomial closures in valuation domains and proves they are not topological in rank at least two, a novel insight into their structural properties.

Findings

Polynomial closure is characterized for pseudo-convergent sequences.

Polynomial closure is not topological in valuation domains of rank ≥ 2.

The result applies to valuation domains of arbitrary rank.

Abstract

We characterize the polynomial closure of a pseudo-convergent sequence in a valuation domain $V$ of arbitrary rank, and then we use this result to show that the polynomial closure is never topological when $V$ has rank at least $2$.