# Singularities of Base Polynomials and Gau-Wu Numbers

## TL;DR

This paper explores how the singularities of base curves influence the Gau-Wu number of matrices, using algebraic geometry, and extends existing classifications to broader matrix cases with new proofs.

## Contribution

It introduces an algebraic geometric approach to classify Gau-Wu numbers based on base curve singularities and extends prior work to unitarily irreducible matrices with reducible base curves.

## Key findings

- Necessary conditions for k(A)=2 in unitarily irreducible matrices.
- Characterization of matrices with k(A)=n.
- Examples showing singularities do not fully determine k(A).

## Abstract

In 2013, Gau and Wu introduced a unitary invariant, denoted by $k(A)$, of an $n\times n$ matrix $A$, which counts the maximal number of orthonormal vectors $\textbf x_j$ such that the scalar products $\langle A\textbf x_j,\textbf x_j\rangle$ lie on the boundary of the numerical range $W(A)$. We refer to $k(A)$ as the Gau--Wu number of the matrix $A$. In this paper we take an algebraic geometric approach and consider the effect of the singularities of the base curve, whose dual is the boundary generating curve, to classify $k(A)$. This continues the work of Wang and Wu classifying the Gau-Wu numbers for $3\times 3$ matrices. Our focus on singularities is inspired by Chien and Nakazato, who classified $W(A)$ for $4\times 4$ unitarily irreducible $A$ with irreducible base curve according to singularities of that curve. When $A$ is a unitarily irreducible $n\times n$ matrix, we give necessary conditions for $k(A) = 2$, characterize $k(A) = n$, and apply these results to the case of unitarily irreducible $4\times 4$ matrices. However, we show that knowledge of the singularities is not sufficient to determine $k(A)$ by giving examples of unitarily irreducible matrices whose base curves have the same types of singularities but different $k(A)$. In addition, we extend Chien and Nakazato's classification to consider unitarily irreducible $A$ with reducible base curve and show that we can find corresponding matrices with identical base curve but different $k(A)$. Finally, we use the recently-proved Lax Conjecture to give a new proof of a theorem of Helton and Spitkovsky, generalizing their result in the process.