Integer-valued Polynomials, Pr\"ufer Domains and the Sacked Bases Property

TL;DR

This paper investigates whether certain Pr"ufer domains, specifically those formed by integer-valued polynomials, have the stacked bases property, providing proofs for specific cases and reducing the problem to matrix questions.

Contribution

It proves that all integer-valued polynomial Pr"ufer domains on rank-one valuation domains have the stacked bases property and reduces the problem for rings of integers of number fields to matrix analysis.

Findings

Integer-valued polynomial Pr"ufer domains on rank-one valuation domains possess the stacked bases property.

The study reduces the stacked bases property question for rings of integers of number fields to 2x2-matrix problems.

Abstract

To study the question of whether every two-dimensional Pr\"ufer domain possesses the stacked bases property, we consider the particular case of the Pr\"ufer domains formed by integer-valued polynomials. The description of the spectrum of the rings of integer-valued polynomials on a subset of a rank-one valuation domain enables us to prove that they all possess the stacked bases property. We also consider integer-valued polynomials on rings of integers of number fields and we reduce in this case the study of the stacked bases property to questions concerning $2\times 2$-matrices.