Twin Vector Fields and Independence of Spectra for Quadratic Vector Fields

Abstract

The object of this paper is to address the following question: When is a polynomial vector field on $\mathbb{C}^2$ completely determined (up to affine equivalence) by the spectra of its singularities? We will see that for quadratic vector fields this is not the case: given a generic quadratic vector field there is, up to affine equivalence, exactly one other vector field which has the same spectra of singularities. Moreover, we will see that we can always assume that both vector fields have the same singular locus and at each singularity both vector fields have the same spectrum. Let us say that two vector fields are twin vector fields if they have the same singular locus and the same spectrum at each singularity. To formalize the above claim we shall prove the following: any two generic quadratic vector fields with the same spectra of singularities (yet possibly different singular locus) can be transformed by suitable affine maps to be either the same vector field or a pair of twin vector fields. We then analyze the case of quadratic Hamiltonian vector fields in more detail and find necessary and sufficient conditions for a collection of non-zero complex numbers to arise as the spectra of singularities of a quadratic Hamiltonian vector field. Lastly, we show that a generic quadratic vector field is completely determined (up to affine equivalence) by the spectra of its singularities together with the characteristic numbers of its singular points at infinity.