A pencil of quadratic forms in nine variables with no member of Witt index four

Abstract

We exhibit an explicit pair $(A,B)$ of integral symmetric $9\times 9$ matrices, defining a nonsingular pair of quadratic forms over $\mathbb{Q}$, such that no member of the rational pencil $\lambda q_A+\mu q_B$ 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 $\det (\lambda A+\mu B)$, 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.