On Families QSL $_{\ge 2}$ of Quadratic Systems with Invariant Lines of Total Multiplicity At Least 2

Abstract

Let QSL $_{\ge i}$ be the family of quadratic differential systems with invariant lines of total multiplicity at least i and let QSL $_{i}$ denote the family of quadratic systems with invariant lines of total multiplicity exactly i. For any polynomial system the line at infinity is invariant. Thus the family ${{\varvec{QS}}}\,$ of all quadratic systems is the same as QSL $_{\ge 1}$. Our main interest is in the family QSL $_{\ge 2}$, the largest strict sub-family of ${{\varvec{QS}}}\,$ of systems having invariant lines. Our first goal is to give a brief survey of what we know so far about this family. At the moment we know everything about the family QSL $_{\ge 4}$ as well as about two important subfamilies of QSL $_{3}$. Only two other subfamilies of QSL $_{3}$ remain to be studied. Our second goal is to give necessary and sufficient conditions in terms of invariant polynomials for systems to belong to one of these, namely the family QSL $^{2p}$ of quadratic systems with two parallel lines real or complex and to give the geometrical classification of this family. Our third goal is to complete a former partial result in the literature and obtain the global result on QSL $_{\ge 2}$ saying that the Hilbert number for this family is one. This opens the road for attempting to obtain the topological classification of QSL $_{\ge 2}$.