Noetherianity and Length of Melnikov Functions

We study foliations in C² C 2 given by polynomial deformations of the form dH+ε η =0 d H + ϵ η = 0 , with γ (t)⊂ H⁻¹(t) γ ( t ) ⊂ H - 1 ( t ) a family of cycles. The Poincaré first return map is of the form P(t)=t+∑ ⱼ ε ʲ Mⱼ^γ (t). P ( t ) = t + ∑ j ϵ j M j γ ( t ) . The functions Mⱼ^γ M j γ are called Melnikov functions and are given by iterated integrals of orbit length at most j . We show that, for each k∈ N k ∈ N , there exists a universal Noetherianity index nH,γ(k) n H , γ ( k ) , independent of the deformation η η , such that, if Mⱼ^γ ≡ 0 M j γ ≡ 0 , for j=1,… ,nH,γ(k) j = 1 , … , n H , γ ( k )