PreprintCC BY 4.0arXiv
Polynomial parametrization of SL2 over global function field rings
Abstract
We combine the universal polynomial matrix identities of Larsen--Nguyen with Trost's arithmetic lemmas to obtain polynomial parametrizations of SL2(𝒪K,S) for every global function field K and every nonempty finite set of places S. The additional ingredient is a function-field cancellation lemma, proved using ray class fields and Chebotarev in odd characteristic and the perfectness of finite residue fields in characteristic two. The proof also gives a polynomial matrix over ℤ that works simultaneously for all these rings.
Related
- On the Krull dimensions of rings of integer-valued polynomials and polynomial rings
- Stacked-bases Properties, maximal ideals and polynomial closures of Rings of Integer-Valued Polynomials
- Cryptanalysis of a Class of Ideal Secret Sharing Schemes Based on the CRT for Polynomial Rings
- Algebraic Structure and Orthogonality Properties of Rational Polynomial Phase Rings on Discrete Lattices
- Universality Criterion Sets for Definite Quadratic Lattices over Polynomial Rings
BibTeX
@misc{nguyen2026polynomial,
title = {{Polynomial parametrization of $\textSL_2$ over global function field rings}},
author = {Dong Quan Ngoc Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12705},
eprint = {2610.12705},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12705},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Polynomial parametrization of SL_2 over global function field rings
AU - Dong Quan Ngoc Nguyen
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12705
UR - https://arxiv.org/abs/2610.12705
AB - We combine the universal polynomial matrix identities of Larsen--Nguyen with Trost's arithmetic lemmas to obtain polynomial parametrizations of SL_2(𝒪_K,S) for every global function field K and every nonempty finite set of places S. The additional ingredient is a function-field cancellation lemma, proved using ray class fields and Chebotarev in odd characteristic and the perfectness of finite residue fields in characteristic two. The proof also gives a polynomial matrix over ℤ that works simultaneously for all these rings.
ER -