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

TL;DR

This work extends non-Archimedean eigenvalue inclusion beyond Gershgorin disks to Brauer’s oval (Cassini) regions for matrices $A\in M_n(\mathbb{K})$ over a non-Archimedean field $\mathbb{K}$. It proves a non-Archimedean Brauer Oval Theorem, giving two Cassini-type inclusions using $h_j(A)=\max_{k\neq j}|a_{j,k}|$ and $v_k(A)=\max_{j\neq k}|a_{j,k}|$, namely $\sigma(A)\subseteq \bigcup_{j\neq k} \{z: |z-a_{j,j}||z-a_{k,k}|\le h_j(A)h_k(A)\}$ and $\sigma(A)\subseteq \bigcup_{j\neq k} \{z: |z-a_{j,j}||z-a_{k,k}|\le v_j(A)v_k(A)\}$. The paper shows these inclusions yield bounds for zeros of polynomials via Frobenius/companion matrices and proves their equivalence to the non-Archimedean Ostrowski nonsingularity theorem of Li–Li (2025). This generalizes the Nica–Sprague disk theorem to oval regions, providing sharper non-Archimedean spectral bounds with potential computational applications in polynomial root finding over non-Archimedean fields.

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}].

Theorems & Definitions (36)

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