二次形式奇异对的伴随闭包与Pfister型准则
Adjoint Closures of Singular Quadratic Pencils and First's Pfister-Type Conjecture
- Nanjing University（南京大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文解决二次形式奇异对的伴随闭包问题，证明非奇异性是弱双曲性的必要条件，构造K^7上的反例并确定机制，证明7是极小维度。
AI中文摘要：
设K为特征不等于2的域。此前已证明，对于数域或实闭域上的非奇异二次形式对，弱双曲性等价于其伴随闭包上的全符号消失，并提出非奇异性是否为必要条件的问题。我们证明该条件确实必要。在每个形式实域上，我们在K^7上构造一个奇异对，其伴随闭包恰好是其二维 pencil（ pencil 译为“束”）且完全由双曲形式组成，尽管该对并非弱双曲。随后我们明确其机制：对于每个正最小指数的对称奇异克罗内克块M_ε，其伴随闭包即为其束；此外，将M_ε附加到非奇异对会将可见正则闭包坍缩为原束。相比之下，最小指数为零的块会保留完整正则闭包。这产生了精确的可见性二分法和一类通用反例。最后，在数域和实闭域上，我们证明维度7是极小的。对于该极小例子，我们还计算了完整伴随代数及其对合迹形式。
英文摘要：
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.
