Adjoint Closures of Singular Quadratic Pencils and First's Pfister-Type Conjecture
Abstract
We construct, over every formally real field, a singular pair of quadratic forms in dimension seven whose adjoint closure equals its pencil and consists entirely of hyperbolic forms in First's sense, although the pair is not weakly hyperbolic. This disproves First's conjecture that his local--global criterion for nonsingular pairs extends to singular pairs. The example has minimal dimension over formally real number fields and real closed fields. The construction uses an explicit closure formula: adjoining a symmetric singular Kronecker block of positive minimal index to a pair with a nondegenerate first member forces the adjoint closure of the sum to equal its pencil. We determine the corresponding closure formulas for Kronecker decompositions and compute the adjoint algebra, its Jacobson radical, and the involution-trace form of the seven-dimensional example.
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BibTeX
@misc{xu2026adjoint,
title = {{Adjoint Closures of Singular Quadratic Pencils and First's Pfister-Type Conjecture}},
author = {Shisong Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.21017},
eprint = {2608.21017},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21017},
}
RIS
TY - UNPB
TI - Adjoint Closures of Singular Quadratic Pencils and First's Pfister-Type Conjecture
AU - Shisong Xu
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2608.21017
UR - https://arxiv.org/abs/2608.21017
AB - We construct, over every formally real field, a singular pair of quadratic forms in dimension seven whose adjoint closure equals its pencil and consists entirely of hyperbolic forms in First's sense, although the pair is not weakly hyperbolic. This disproves First's conjecture that his local--global criterion for nonsingular pairs extends to singular pairs. The example has minimal dimension over formally real number fields and real closed fields. The construction uses an explicit closure formula: adjoining a symmetric singular Kronecker block of positive minimal index to a pair with a nondegenerate first member forces the adjoint closure of the sum to equal its pencil. We determine the corresponding closure formulas for Kronecker decompositions and compute the adjoint algebra, its Jacobson radical, and the involution-trace form of the seven-dimensional example.
ER -