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