The Weierstrass preparation theorem and resultants of $p$-adic power series

TL;DR

This paper introduces a new definition of the resultant for $p$-adic power series by establishing a universal version of the Weierstrass preparation theorem, advancing the understanding of $p$-adic analytic functions.

Contribution

It provides a novel definition of the resultant for $p$-adic power series based on a universal Weierstrass preparation theorem, extending classical concepts to the $p$-adic setting.

Findings

Defined the resultant of two $p$-adic power series.

Proved a universal version of the Weierstrass preparation theorem.

Enabled new approaches to $p$-adic analytic functions.

Abstract

We define the resultant of two power series with coefficients in the ring of integers of a $p$-adic field. In order to do this, we prove a universal version of the Weierstrass preparation theorem.