# Characteristic polynomials of p-adic matrices

## TL;DR

This paper investigates the precision of characteristic polynomials of p-adic matrices, providing criteria and algorithms to determine and optimize their precision based on differential methods.

## Contribution

It introduces a checkable criterion for the exact precision of characteristic polynomials and an efficient algorithm to determine optimal precision when the criterion fails.

## Key findings

- A criterion for the exact precision of characteristic polynomials in p-adic matrices.
- An O~(n^3) algorithm for computing optimal precision.
- Examples showing cases of higher-than-expected precision.

## Abstract

We analyze the precision of the characteristic polynomial of an $n\times n$ p-adic matrix A using differential precision methods developed previously. When A is integral with precision O(p^N), we give a criterion (checkable in time O~(n^omega)) for $\chi$(A) to have precision exactly O(p^N). We also give a O~(n^3) algorithm for determining the optimal precision when the criterion is not satisfied, and give examples when the precision is larger than O(p^N).