AI 中文总结
本研究针对素数模阶乘剩余集的大小下界问题，利用有限域阶乘恒等式、分式线性映射交点性质与Stevens-de Zeeuw点线相交定理，将下界从平方根级改进至\\(p^{8/15}\\)量级。
AI 中文摘要
对于素数\\(p\\)，设\\(A_p=\{k!\pmod p:1\leq k<p\}\\)。我们证明了\\(|A_p|\gg p^{8/15}\\)，改进了通用下界\\((\sqrt{2}-o(1))p^{1/2}\\)。证明从有限域\\(\mathbb{F}_p\\)中的恒等式\\((n+2)!=(n+1)!+((n+1)!)^2/n!\\)展开，该恒等式为一族分式线性映射生成大量交点。经柯西-施瓦茨不等式处理后，该族中两个映射间的转移映射变为重数至多为2的仿射直线。随后利用Stevens和de Zeeuw的笛卡尔积点线相交定理得到指数8/15。
英文摘要
For a prime \(p\), let \(A_p=\{k!\pmod p:1\leq k<p\}\). We prove \(|A_p|\gg p^{8/15}\), improving the general lower bound \((\sqrt{2}-o(1))p^{1/2}\). The proof begins with the identity \((n+2)!=(n+1)!+((n+1)!)^2/n!\) in \(\mathbb{F}_p\), which produces many incidences for a family of fractional-linear maps. After Cauchy--Schwarz, the transition maps between two members of this family become affine lines, with multiplicity at most two. The Cartesian-product point-line incidence theorem of Stevens and de Zeeuw then yields the exponent \(8/15\).
Comments6 pages, 1 figure. Comments welcome
