About the solvability of matrix polynomial equations

TL;DR

This paper investigates the conditions under which self-adjoint matrix polynomial equations, especially of odd degree with non-degenerate leading form, have solutions, expanding understanding of their solvability.

Contribution

It proves that odd-degree self-adjoint matrix polynomial equations with non-degenerate leading form always have self-adjoint solutions, and explores solutions for even degree and multivariable cases.

Findings

Self-adjoint solutions exist for odd-degree equations with non-degenerate leading form.

Conditions for solvability of even-degree and multivariable equations are analyzed.

Main result guarantees solutions under specific algebraic conditions.

Abstract

We study self-adjoint matrix polynomial equations in a single variable and prove existence of self-adjoint solutions under some assumptions on the leading form. Our main result is that any self-adjoint matrix polynomial equation of odd degree with non-degenerate leading form can be solved in self-adjoint matrices. We also study equations of even degree and equations in many variables.