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

Nica and Sprague [$\mathit{\text{Am.}}\mathit{\text{Math.}}\mathit{\text{Mon.,}}\mathit{\text{2023}}$] derived a non-Archimedean version of the Gershgorin disk theorem. We derive a non-Archimedean version of the oval (of Cassini) theorem by Brauer [$\mathit{\text{Duke}}\mathit{\text{Math.}}\mathit{\text{J.,}}\mathit{\text{1947}}$] 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 [$\mathit{\text{J.}}\mathit{\text{Comput.}}\mathit{\text{Appl.}}\mathit{\text{Math.,}}\mathit{\text{2025}}$].