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

TL;DR

This paper calculates the density of p-adic polynomials that generate extensions with a fixed splitting type, providing rational functions, recursive formulas, and asymptotic behavior in the tame case.

Contribution

It introduces explicit formulas and recursive methods for computing densities of polynomials with specified splitting types over local fields.

Findings

Density is a rational function of residue field size in the tame case.

Provides a recursive formula for these densities.

Analyzes asymptotic behavior as residue field size grows.

Abstract

We prove that the density of polynomials $P(x)=\sum_{i=0}^n a_n x^n$ over a local field $K$ generating an \'etale 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.