In this talk I will present some very recent results obtained in collaboration with Yury Kudryashov. These results are a continuation of those I presented last semester in this seminar (Nov. 11: "Polynomial vector fields on $\mathbb{C}^2$ and their spectra of singularities").

Consider a polynomial vector field of degree $n$ on $\mathbb{C}^2$, having an invariant line at infinity and isolated singularities only. The extended spectra of singularities is the collection of the spectra of the linearization matrices of each singular point on the affine part, together with the characteristic numbers at infinity. This collection consists of $2n^2+n+1$ complex numbers and is invariant under affine equivalence of vector fields.

These numbers are related by 4 classical index theorems: the Euler-Jacobi relations, the Baum-Bott theorem, and the Camacho-Sad theorem. However, a dimensional count shows that there must exist yet more algebraic relations among these numbers. The question is: what are those "hidden relations"?

In this talk we will answer the question in the quadratic case, where there is only one such missing relation. Moreover, we will see that this equation does not come from an index theorem. In fact, we can show that the only "index-theorem like" equations relating the extended spectra come from the four classical index theorems mentioned above.