# Algebraic condition for the singularity of certain T\"oplitz pencils

## TL;DR

This paper presents an algebraic condition based on principal minors to determine the singularity of specific T"oplitz matrix pencils, linking it to the T"oplitz pencil conjecture.

## Contribution

It introduces a new algebraic criterion involving principal minors for T"oplitz pencil singularity, connecting to an existing conjecture.

## Key findings

- Derived an algebraic condition for T"oplitz pencil singularity
- Established equivalence with the T"oplitz pencil conjecture
- Proposed an algebraic conjecture related to matrix minors

## Abstract

An algebraic condition for the singularity of certain T\"oplitz matrix pencils is derived which involves only the principal minors of the constant parts of the pencils. This leads to an algebraic conjecture which is equivalent to the so-called T\"oplitz pencil conjecture.