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Stacked-bases Properties, maximal ideals and polynomial closures of Rings of Integer-Valued Polynomials
Abstract
We study the integer-valued polynomial ring with the stacked-bases property, maximal ideals above valuation ideals, and the polynomial closure of matrices with prescribed characteristic polynomials. A content argument gives stacked-bases properties for integer-valued polynomial rings over number-field integer rings. The construction of Hagg and Mortberg provides a Bezout domain without this property. For rank-one valuation domains, maximal ideals are described by evaluations and pseudo-convergent sequences. For matrices, we give congruence-ideal and companion-lattice criteria, together with finite-level tests to characterize the polynomial closures.
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BibTeX
@misc{chen2026stackedbases,
title = {{Stacked-bases Properties, maximal ideals and polynomial closures of Rings of Integer-Valued Polynomials}},
author = {Xuyang Chen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00906},
eprint = {2611.00906},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00906},
}
RIS
TY - UNPB
TI - Stacked-bases Properties, maximal ideals and polynomial closures of Rings of Integer-Valued Polynomials
AU - Xuyang Chen
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00906
UR - https://arxiv.org/abs/2611.00906
AB - We study the integer-valued polynomial ring with the stacked-bases property, maximal ideals above valuation ideals, and the polynomial closure of matrices with prescribed characteristic polynomials. A content argument gives stacked-bases properties for integer-valued polynomial rings over number-field integer rings. The construction of Hagg and Mortberg provides a Bezout domain without this property. For rank-one valuation domains, maximal ideals are described by evaluations and pseudo-convergent sequences. For matrices, we give congruence-ideal and companion-lattice criteria, together with finite-level tests to characterize the polynomial closures.
ER -