PreprintCC BY-NC-ND 4.0arXiv
Universality Criterion Sets for Definite Quadratic Lattices over Polynomial Rings
Abstract
Let k be a field of characteristic ≠ 2. Under suitable assumptions on the dimensions of anisotropic quadratic forms over k and k(x), we show that a definite integral quadratic form over the polynomial ring k[x] is universal if the form represents all scalars from k and all scalar multiples of x. The coefficient field k in our main result can be a finite field, a non-dyadic higher local field, a global function field, or a one-variable function field over ℂ or ℂ(t). We have a finite universality criterion set over k[x] for some coefficient fields k, including some cases where k has infinitely many square classes (e.g., k=ℂ(t)).
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BibTeX
@misc{hu2026universality,
title = {{Universality Criterion Sets for Definite Quadratic Lattices over Polynomial Rings}},
author = {Yong Hu and Jing Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37339},
eprint = {2610.37339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37339},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Universality Criterion Sets for Definite Quadratic Lattices over Polynomial Rings
AU - Yong Hu
AU - Jing Liu
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37339
UR - https://arxiv.org/abs/2610.37339
AB - Let k be a field of characteristic ≠ 2. Under suitable assumptions on the dimensions of anisotropic quadratic forms over k and k(x), we show that a definite integral quadratic form over the polynomial ring k[x] is universal if the form represents all scalars from k and all scalar multiples of x. The coefficient field k in our main result can be a finite field, a non-dyadic higher local field, a global function field, or a one-variable function field over ℂ or ℂ(t). We have a finite universality criterion set over k[x] for some coefficient fields k, including some cases where k has infinitely many square classes (e.g., k=ℂ(t)).
ER -