Affine monodromy and exact value distributions over finite fields

Abstract

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.