Noetherianity and Length of Melnikov Functions

Source abstract

Abstract We study foliations in $$\mathbb {C}^2$$ C 2 given by polynomial deformations of the form $$dH+\epsilon \eta =0$$ d H + ϵ η = 0 , with $$\gamma (t)\subset H^{-1}(t)$$ γ ( t ) ⊂ H - 1 ( t ) a family of cycles. The Poincaré first return map is of the form $$P(t)=t+\sum _j \epsilon ^j M_j^\gamma (t).$$ P ( t ) = t + ∑ j ϵ j M j γ ( t ) . The functions $$M_j^\gamma$$ M j γ are called Melnikov functions and are given by iterated integrals of orbit length at most j . We show that, for each $$k\in \mathbb {N}$$ k ∈ N , there exists a universal Noetherianity index $$n_{\scriptscriptstyle H,\gamma }(k)$$ n H , γ ( k ) , independent of the deformation $$\eta$$ η , such that, if $$M_j^\gamma \equiv 0$$ M j γ ≡ 0 , for $$j=1,\ldots ,n_{ H,\gamma }(k)$$ j = 1 , … , n H , γ ( k ) , then $$M_j^\gamma$$ M j γ is of orbit length $$j-k$$ j - k , for any Melnikov function $$M_j^\gamma$$ M j γ . We call the smallest index with this property just the Noetherianity index $$\nu _{\scriptscriptstyle H,\gamma }(k)$$ ν H , γ ( k ) . To prove this theorem, we develop a structure theorem for Melnikov functions and use the Ritt–Raudenbush differential algebra theorem. We calculate the universal Noetherianity index $$n_{H,\gamma }(k)$$ n H , γ ( k ) in various nontrivial examples.