Division and Slope Factorization of p-Adic Polynomials

TL;DR

This paper introduces an efficient algorithm for slope factorization of p-adic polynomials over complete discrete valuation fields, emphasizing stability and avoiding fractional exponents.

Contribution

It presents a novel Newton iteration-based method for slope factorization that improves efficiency and stability without fractional exponents.

Findings

Algorithm achieves stable slope factorizations efficiently.

Avoids fractional exponents in computations.

Analyzes behavior under various precision models.

Abstract

We study two important operations on polynomials defined over complete discrete valuation fields: Euclidean division and factorization. In particular, we design a simple and efficient algorithm for computing slope factorizations, based on Newton iteration. One of its main features is that we avoid working with fractional exponents. We pay particular attention to stability, and analyze the behavior of the algorithm using several precision models.