Nilpotent Systems Admitting an Algebraic Inverse Integrating Factor over $${{\mathbb{C}}((x,y))}$$

Abstract

We characterize the nilpotent systems whose lowest degree quasihomogeneous term is (y, σ x n ) T , σ = ±1, which have an algebraic inverse integrating factor over C ((x, y)). In such cases, we show that the systems admit an inverse integrating factor of the form (h +· · · ) q with h = 2σ x n+1 −(n +1)y 2 and q a rational number. We analyze its uniqueness modulus a multiplicative constant.