Non-Archimedean Brauer Oval (of Cassini) Theorem and Applications

TL;DR

This paper extends classical theorems about matrix eigenvalues and polynomial zeros into the non-Archimedean setting, providing new bounds and equivalences relevant for non-Archimedean fields.

Contribution

It derives a non-Archimedean version of Brauer's oval theorem, generalizing previous disk theorems and establishing equivalence with a non-Archimedean Ostrowski theorem.

Findings

Non-Archimedean oval theorem derived

Applications for polynomial zero bounds provided

Equivalence with non-Archimedean Ostrowski theorem established

Abstract

Nica and Sprague [\textit{Am. Math. Mon., 2023}] derived a non-Archimedean version of the Gershgorin disk theorem. We derive a non-Archimedean version of the oval (of Cassini) theorem by Brauer [\textit{Duke Math. J., 1947}] which generalizes the Nica-Sprague disk theorem. We provide applications for bounding the zeros of polynomials over non-Archimedean fields. We also show that our result is equivalent to the non-Archimedean version of the Ostrowski nonsingularity theorem derived by Li and Li [\textit{J. Comput. Appl. Math., 2025}].