Gröbner Bases of algebraic invariants in polynomial differential systems

We consider polynomial differential systems with coefficients in a field K of characteristic zero and the action of a given group G of lin-ear transformations on these systems. The fundamental idea is based on the test of membership in an ideal. We use Gröbner bases to describe the algebra of the covariants in relation to this group of transformations while developing an algorithmic method to construct a minimal system of generators of the covariants of these systems. We are able to determine whether any polynomial expression in the coefficients of the given differ-ential system and the contravariant vector x is a G-covariant and we have an algorithm to calculate its normal form which can be used to express the invariants in a given basis of the G-covariants of these systems. We will give an application in the case of quadratic differential systems when G is the group of centro-affine transformations GL(2,R). 1. Motivations One of the tools used to study polynomial differential systems is the alge-braic invariants under the action of a group of linear transformations. It would be then interesting to establish invariant conditions, in relation to the given group of transformations, that give the existence and the nature of singular points, an