Determinantal Analysis of the Polynomial Integrability of Differential Systems

This work deals with the polynomial and formal (formal series) integrability of the polynomial differential systems around a singular point, namely the conditions which assure the start of the algorithmic process for computing the polynomial or the formal first integrals. When the linear part of the differential system is nonzero, we have established ([9]) the existence of the so called starting equations whose (integer) solutions are exactly the partition of the lower degree of the eventual formal first integrals.
In this work, we study some extensions of the starting equations to the case when the linear part is zero and, particularly, to the bidimensionnal homogeneous differential systems. The principal tool used here is the classical invariant theory.