On geometry of p-adic polynomials

TL;DR

This paper explores a p-adic analogue of the Gauss-Lucas theorem, analyzing the geometric properties of polynomials over the algebraic closure of p-adic numbers.

Contribution

It introduces a novel geometric framework for understanding p-adic polynomials, extending classical complex polynomial results to the p-adic setting.

Findings

Established a p-adic version of the Gauss-Lucas theorem

Described geometric properties of p-adic polynomial roots

Extended classical polynomial theorems to p-adic context

Abstract

An analogue of the Gauss-Lucas theorem for polynomials over the algebraic closure $\mathbb C_p$ of the field of $p$-adic numbers is considered.