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arXiv · 2610.37339

Universality Criterion Sets for Definite Quadratic Lattices over Polynomial Rings

Yong Hu·Jing Liu
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Abstract

Let $k$ be a field of characteristic $\neq 2$. 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 $\mathbb{C}$ or $\mathbb{C}(t)$. 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., $k=\mathbb{C}(t)$).

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Yong Hu, Jing Liu. 2026-10-27. Universality Criterion Sets for Definite Quadratic Lattices over Polynomial Rings. https://arxiv.org/abs/2610.37339

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