AI 中文总结
研究通过单值群不动点统计有限域值分布。给出仿射单值性最优机制的多项式实现，计算特定多项式的纤维枚举器及与上界的关系，包括全仿射情况在不同有限域上的情况及缺陷。
AI 中文摘要
我们通过单值群的不动点统计来研究有限域值分布。对于正则\(n\)次覆盖，遗漏值由错排控制。自然对称单值性给出支撑密度\(1 - D_n / n! \to 1 - e^{-1}\)，而卡梅伦 - 科恩界给出通用上限\(1 - 1 / n\)，由强\(2 -\)传递仿射单值性达到。我们给出了这种最优机制的显式多项式实现。对于\(N = p^e\)且\(h\mid N - 1\)，设置\(\Lambda_{N,h}(U)=U\bigl(U^{(N - 1)/h}-1\bigr)^h\)。其几何伽罗瓦闭包是有理的，\(U = z^h\)，\(T=(z^N - z)^h\)，其几何单值性是仿射群\((\F_N, +)\rtimes H_h\)。对于每个\(\F_Q / \F_p\)扩展，我们精确计算\(\Lambda_{N,h}\)的完整纤维枚举器，包括非正则情况。在全仿射情况\(h = N - 1\)时，多项式\(U(U - 1)^{N - 1}\)在每个包含\(\F_N\)的有限域上对于非置换多项式达到万 - 石尾 - 陈上界；在任意扩展上，我们计算该界的精确缺陷。
英文摘要
We study finite field value distributions through the fixed-point statistics of monodromy groups. For a regular degree-\(n\) cover, omitted values are controlled by derangements. Thus natural symmetric monodromy gives support density \(1-D_n/n!\to 1-e^{-1}\), while the Cameron--Cohen bound gives the universal ceiling \(1-1/n\), attained by sharply \(2\)-transitive affine monodromy. We give an explicit polynomial realization of this optimal mechanism. For \(N=p^e\) and \(h\mid N-1\), set \[ Λ_{N,h}(U)=U\bigl(U^{(N-1)/h}-1\bigr)^h . \] Its geometric Galois closure is rational, \[ U=z^h, \qquad T=(z^N-z)^h, \] and its geometric monodromy is the affine group \((\F_N,+)\rtimes H_h\). For every extension \(\F_Q/\F_p\) we compute the complete fibre enumerator of \(Λ_{N,h}\) exactly, including the nonregular cases. In the full affine case \(h=N-1\), the polynomial \[ U(U-1)^{N-1} \] attains the Wan--Shiue--Chen upper bound for non-permutation polynomials over every finite field containing \(\F_N\); over arbitrary extensions we compute the exact defect from that bound.
Comments26 pages
