Density of p-adic polynomials generating extensions with fixed splitting type

2022/11/18 by John Yin, Yin, John
Mathematics · #advanced mathematical theories #Mathematical Dynamics and Fractals #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2211.10425

Abstract

We prove that the density of polynomials P(x)=∑_i=0ⁿ a_n xⁿ over a local field K generating an étale extension with specified splitting type is a rational function in terms of the size of the residue field of K in the case where the splitting type is tame. Moreover, we give a computable recursive formula for these densities and compute the asymptotics of this density as the size of the residue field tends to infinity.

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