Spectra of quadratic vector fields on $\mathbb{C}^2$: The missing relation

TL;DR

This paper investigates the invariant spectra of quadratic vector fields on a2^2, revealing five key polynomial relations among these spectral invariants, including a newly discovered relation that cannot be derived from classical index theorems.

Contribution

It explicitly identifies and formulates the previously unknown fifth polynomial relation among the spectra of singularities in quadratic vector fields.

Findings

Identified five polynomial relations among spectral invariants.

Discovered and explicitly formulated the missing fifth relation.

Proved the fifth relation cannot be derived from existing index theorems.

Abstract

Consider a quadratic vector field on $\mathbb{C}^2$ having an invariant line at infinity and isolated singularities only. We define the extended spectra of singularities to be the collection of the spectra of the linearization matrices of each of the singular points over the affine part, together with all the characteristic numbers (i.e. Camacho-Sad indices) at infinity. This collection consists of 11 complex numbers, and is invariant under affine equivalence of vector fields. In this paper we describe all polynomial relations among these numbers. There are 5 independent polynomial relations; four of them follow from the Euler-Jacobi, the Baum-Bott and the Camacho-Sad index theorems, and are well known. The fifth relation was, until now, completely unknown. We provide an explicit formula for the missing 5th relation, discuss it's meaning and prove that it cannot be formulated as an index theorem.