The 8-universality Criterion is Unique

Uniqueness of the 2-universality Criterion

Kim, Kim, and Oh gave a minimal criterion for the 2-universality of positive-definite integer-matrix quadratic forms. We show that this 2-universality criterion is unique in the sense of the uniqueness of the Conway-Schneeberger Fifteen Theorem.

Minimal universality criterion sets on the representations of quadratic forms

For a set $S$ of (positive definite and integral) quadratic forms with bounded rank, a quadratic form $f$ is called $S$-universal if it represents all quadratic forms in $S$. A subset $S_0$ of $S$ is called an $S$-universality criterion set if any $S_0$-universal quadratic form is $S$-universal. We say $S_0$ is minimal if there does not exist a proper subset of $S_0$ that is an $S$-universality criterion set. In this article, we study various properties of minimal universality criterion sets. In particular, we show that for `most' binary quadratic forms $f$, minimal $S$-universality criterion …

数域上二次型的泛准则集

Universality criterion sets for quadratic forms over number fields

本文通过最小准则集的新刻画，证明在完全实代数数域上，二次型表示所有完全正整数的有限准则集总是存在且唯一，并推广到给定整数子集的表示。

具有最小判定集的 $m$ 角型的紧致普适性

Tight universality of $m$-gonal forms with minimal criterion sets

本文确定了紧致 $\mathcal{T}(n)$-普适 $m$ 角型的最小判定集，给出了 $n\geq8$ 时除两个例外情形的完整刻画，并证明 $m$ 的上界最优。

实二次域上的 escalations 与判别准则

Escalations and criteria over real quadratic fields

针对全实代数数域上通用二次型的判别准则仅在Q(√5)上明确已知的问题，本文开发数域上的escalate方法，证明判别准则有限性并提供计算工具，对Q(√2)、Q(√3)进行计算得到猜想，还发展对角二次型的escalation理论并给出对应结果。

A quick route to unique factorization in quadratic orders

We give a short proof -- not relying on ideal classes or the geometry of numbers -- of a known criterion for quadratic orders to possess unique factorization.