Local Descriptions of Roots of Cubic Equations over P-adic Fields

TL;DR

This paper provides a detailed analysis of the local behavior of roots of cubic equations over p-adic fields for primes greater than 3, addressing a longstanding open question in p-adic polynomial root localization.

Contribution

It offers new local descriptions of roots of cubic equations over p-adic fields, filling a gap in understanding for primes greater than 3.

Findings

Characterization of root domains in p-adic fields

Criteria for roots belonging to specific p-adic subsets

Extension of root localization to cubic equations

Abstract

The most frequently asked question in the $p-$adic lattice models of statistical mechanics is that whether a root of a polynomial equation belongs to domains $\mathbb{Z}_p^{*}, \ \mathbb{Z}_p\setminus\mathbb{Z}_p^{*}, \ \mathbb{Z}_p, \ \mathbb{Q}_p\setminus\mathbb{Z}_p^{*}, \ \mathbb{Q}_p\setminus\left(\mathbb{Z}_p\setminus\mathbb{Z}_p^{*}\right), \ \mathbb{Q}_p\setminus\mathbb{Z}_p, \ \mathbb{Q}_p $ or not. However, this question was open even for lower degree polynomial equations. In this paper, we give local descriptions of roots of cubic equations over the $p-$adic fields for $p>3$.