实二次域上的 escalations 与判别准则
Escalations and criteria over real quadratic fields
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中文总结 AI 辅助
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。
中文摘要 AI 辅助
著名的15定理和290定理完全刻画了有理数域$\boldsymbol{\text{Q}}$上的通用二次型。每个全实代数数域都存在类似的定理，但仅在$\boldsymbol{\text{Q}}(\boldsymbol{\text{√5}})$上明确已知其判别准则集合。我们从理论和计算两方面研究判别准则集合：开发了数域上的 escalate 方法，从而给出了判别准则有限性的简洁证明，更重要的是，提供了计算它们的实用工具。我们通过对$\boldsymbol{\text{Q}}(\boldsymbol{\text{√2}})$和$\boldsymbol{\text{Q}}(\boldsymbol{\text{√3}})$的显式计算对此进行说明，得到了关于对应通用性判别准则的猜想；还证明了许多 escalator 格的通用性。此外，我们开发了针对对角二次型的略有不同的 escalate 理论，得到了$\boldsymbol{\text{Q}}(\boldsymbol{\text{√5}})$的对角判别准则集合，并猜想$\boldsymbol{\text{Q}}(\boldsymbol{\text{√2}})$和$\boldsymbol{\text{Q}}(\boldsymbol{\text{√3}})$也满足该结果。
英文摘要
The famous 15-Theorem and 290-Theorem fully characterise universal quadratic forms over $\mathbb{Q}$. Similar theorems exist for every totally real number field, but only over $\mathbb{Q}(\sqrt5)$ the criterion set is explicitly known. We study criterion sets both theoretically and computationally: We develop the method of escalation over number fields, thus providing a simple proof of finiteness of the criteria and, more importantly, a practical tool for computing them. We illustrate this by explicit computations for $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt3)$, obtaining a conjecture about the corresponding universality criteria; we also prove universality of many escalator lattices. Moreover, we develop the somewhat different theory of escalations for diagonal quadratic forms, obtaining the diagonal criterion set for $\mathbb{Q}(\sqrt5)$ and conjecturally for $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt3)$.
