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

Consider a quadratic vector field on $\mathbb{C}^2$ having an invariant line at infinity and isolated, non-degenerate singularities only. We define the extended spectra of singularities to be the collection of the spectra of the linearization matrices of each singular point 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.