Minimal polynomial basis of GL (2, R )-comitants and of GL (2, R )-invariants of the planar system of differential equations with nonlinearities of the fourth degree

Abstract

The theory of algebraic invariants and comitants for polynomial autonomous systems of differential equations has been developed by C. Sibirschi and his disciples. One of the important problems concerning this theory is the construction of minimal polynomial bases of the invariants and comitants of the mentioned systems, with respect to different subgroups of the affine group of the transformations of their phase planes, in particular with respect to the subgroup GL(2, R). Some important results in this direction are obtained by academician C. Sibirschi and N.Vulpe. We remark, that polynomial bases for different combinations of homogeneous polynomials Pmj (x 1 , x 2) (j = 1, 2, m = 0, 1, 2, 3) in system were considered by E. Gasinskaya-Kirnitskaya, Dang Dinh Bich, D. Boularas, M. Popa, V. Ciobanu, V. Danilyuk, E. Naidenova. We establishe a conjecture that the minimal polynomial basis of GL(2, R)-comitants (respectively, of GL(2, R)-invariants) of system (1) consists from 419 elements (respectively, 182 elements) which must be of the above 111 (respectively, 42) types.