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

  • Tony Quertier
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Abstract

We exhibit an explicit pair (A,B) of integral symmetric 9×9 matrices, defining a nonsingular pair of quadratic forms over ℚ, such that no member of the rational pencil λqA+μqB has Witt index $4$ over 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 ⌈ (n−1)/2⌉. The obstruction is purely $2$-adic and affects the whole pencil at once: every member has Witt index exactly $3$ over ℚ2. The proof is finite and elementary: a parity argument on det(λA+μB), a congruence lemma reducing ℙ1(ℚ2) to the twelve classes of ℙ1(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.

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Cite

BibTeXRIS
BibTeX
@misc{quertier2026pencil,
  title = {{A pencil of quadratic forms in nine variables with no member of Witt index four}},
  author = {Tony Quertier},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07692},
  eprint = {2609.07692},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07692},
}
RIS
TY  - UNPB
TI  - A pencil of quadratic forms in nine variables with no member of Witt index four
AU  - Tony Quertier
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07692
UR  - https://arxiv.org/abs/2609.07692
AB  - We exhibit an explicit pair (A,B) of integral symmetric 9×9 matrices, defining a nonsingular pair of quadratic forms over ℚ, such that no member of the rational pencil λq_A+μq_B has Witt index $4$ over 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 ⌈ (n−1)/2⌉. The obstruction is purely $2$-adic and affects the whole pencil at once: every member has Witt index exactly $3$ over ℚ_2. The proof is finite and elementary: a parity argument on det(λA+μB), a congruence lemma reducing ℙ^1(ℚ_2) to the twelve classes of ℙ^1(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.
ER  -