One subfamily of cubic systems with invariant lines of total multiplicity eight and with two distinct real infinite singularities

Cristina Bujac

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2015), pp. 48-86

Résumé

In this article we classify a subfamily of differential real cubic systems possessing eight invariant straight lines, including the line at infinity and including their multiplicities. This subfamily of systems is characterized by the existence of two distinct infinite singularities, defined by the linear factors of the polynomial $C_3(x,y)=yp_3(x,y)-xq_3(x,y)$, where $p_3$ and $q_3$ are the cubic homogeneities of these systems. Moreover we impose additional conditions related with the existence of triplets and/or couples of parallel invariant lines. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of affine invariant polynomials. The invariant polynomials allow one to verify for any given real cubic system whether or not it has invariant straight lines of total multiplicity eight, and to specify its configuration of straight lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer and the results can therefore be applied for any family of cubic systems in this class, given in any normal form.