AI 中文总结
本文证明了对 $0\le r\le3$，额外特殊群 $E_2$ 与 $C_3^r$ 的直积的小 Davenport 常数为 $2r+10$，通过辛空间中的有限命题和压缩论证给出上界，并留下 $r\ge4$ 的开放问题。
AI 中文摘要
设 $E_2$ 为阶为 $3^5$、指数为三的额外特殊群。我们证明对于 $0\le r\le3$，有 $d(E_2\times C_3^r)=2r+10$。上界由带符号的零块恒等式以及 $\mathbb F_3$ 上四维辛空间中的两个有限命题得出。第一个命题为任意至多十六个非零向量的索引列表上所有可完成的平衡三角形提供边权。第二个命题为不含中心项的关键列表中可能出现的方向族提供权。我们给出完整的覆盖论证、精确的证书文件以及独立实现的检验程序。一个压缩论证消除了在十六项潜在定理中对较小列表长度进行归纳的需要。$r\ge4$ 时的公式仍然开放。
英文摘要
Let $E_2$ be the extraspecial group of order $3^5$ and exponent three. We prove that $d(E_2\times C_3^r)=2r+10$ for $0\le r\le3$. The upper bounds follow from signed zero-block identities and two finite statements in the four-dimensional symplectic space over $\mathbb F_3$. The first supplies edge weights for all completable balanced triangles on any indexed list of at most sixteen nonzero vectors. The second supplies weights for the direction families that can occur in a critical list with no central terms. We give complete coverage arguments, exact certificate files, and separately implemented checking programs. A compression argument removes any need for an induction through smaller list lengths in the sixteen-term potential theorem. The formula for $r\ge4$ remains open.
Comments11 pages; computer-assisted proof; source code, exact certificates, and verification programs included as ancillary files
