发表机构
Orange Research（橙研）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了一个九个变量的非奇异二次型对，其有理束中所有成员在$\mathbb{Q}$上Witt指标均为3，从而反驳了关于Witt指标$\lceil (n-1)/2\rceil$的猜想，证明基于2-adic奇偶性论证和有限类验证。
AI 中文摘要
我们给出一个显式的整数对称$9\times9$矩阵对$(A,B)$，它在$\mathbb{Q}$上定义了一个非奇异的二次型对，使得有理束$\lambda q_A+\mu q_B$中没有任何成员在$\mathbb{Q}$上具有Witt指标$4$。这反驳了\cite{Que16a}中的一个猜想，该猜想预测$n$个变量中的每个非奇异对都会生成一个包含Witt指标为$\lceil (n-1)/2\rceil$的型的束。该障碍纯粹是$2$-进性质的，并且同时影响整个束：每个成员在$\mathbb{Q}_2$上的Witt指标恰好为$3$。证明是有限且初等的：对$\det(\lambda A+\mu B)$进行奇偶性论证，一个同余引理将$\mathbb{P}^1(\mathbb{Q}_2)$约化为$\mathbb{P}^1(\mathbb{Z}/8)$的十二个类，并在每个类上进行验证，其中各向异性判定以两种独立方式得到证实。所有脚本均可在GitHub仓库中获取。
英文摘要
We exhibit an explicit pair $(A,B)$ of integral symmetric $9\times9$ matrices, defining a nonsingular pair of quadratic forms over $\mathbb{Q}$, such that no member of the rational pencil $λq_A+μq_B$ has Witt index $4$ over $\mathbb{Q}$. This refutes a conjecture from \cite{Que16a}, which predicted that every nonsingular pair in $n$ variables generates a pencil containing a form of Witt index $\lceil (n-1)/2\rceil$. The obstruction is purely $2$-adic and affects the whole pencil at once: every member has Witt index exactly $3$ over $\mathbb{Q}_2$. The proof is finite and elementary: a parity argument on $\det(λA+μB)$, a congruence lemma reducing $\mathbb{P}^1(\mathbb{Q}_2)$ to the twelve classes of $\mathbb{P}^1(\mathbb{Z}/8)$, and a verification at each class, in which the anisotropy verdict is certified in two independent ways. All scripts are provided in the GitHub repository.
