PreprintCC BY 4.0arXiv
On the Krull dimensions of rings of integer-valued polynomials and polynomial rings
Abstract
It was conjectured that every integral domain D satisfies dimInt(D) ≤ dim D[X]. We disprove this conjecture and, more strongly, classify all pairs for which both dimensions are finite. Writing m=dim D[X] and n=dimInt(D), the pairs that occur are precisely {(1,1),(2,2)} ∪ {(m,n)∈ℤ>02:m≥3, n≥ m−1}.
Related
- Stacked-bases Properties, maximal ideals and polynomial closures of Rings of Integer-Valued Polynomials
- Integer-valued polynomials over structural matrix rings
- Class groups of polynomial Krull domains
- Localisation and delocalisation of the integer-valued membrane model
- On the probability that the integer parts of polynomials are coprime
BibTeX
@misc{tran2026krull,
title = {{On the Krull dimensions of rings of integer-valued polynomials and polynomial rings}},
author = {Viet-Hoang Tran and Tan M. Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00115},
eprint = {2610.00115},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00115},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Krull dimensions of rings of integer-valued polynomials and polynomial rings
AU - Viet-Hoang Tran
AU - Tan M. Nguyen
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00115
UR - https://arxiv.org/abs/2610.00115
AB - It was conjectured that every integral domain D satisfies dimInt(D) ≤ dim D[X]. We disprove this conjecture and, more strongly, classify all pairs for which both dimensions are finite. Writing m=dim D[X] and n=dimInt(D), the pairs that occur are precisely {(1,1),(2,2)} ∪ {(m,n)∈ℤ_>0^2:m≥3, n≥ m−1}.
ER -