On the saddle order of polynomial differential systems at a resonant singular point
For any system \[ \dot x = px +P(x,y), \quad \dot y= -q y + Q(x,y), \eqno{(1)} \] where \(P\) and \(Q\) are assumed to be complex polynomials of degree \(n\) without linear and constant terms, and \(p\) and \(q\) are positive integers with \(GCD(p,q)=1\), there exists a formal power series \(H(x,y) \) satisfying \(\dot H=\sum_{k\geq 1} L_k (x^q y^p)^{k+1}\). The quantity \(L_k\) is called the \(k\)-th saddle value of system (1); the saddle order of the system is the minimal number \(k\) such that \(L_k\neq 0\). It is proved that for system (1) with \(p=1\) and \(q=2\) for any \(n\geq 2\) such that either \(n+2\) or \(2 n+1\) is a prime number, there always exist polynomials \(P\) and \(Q\) of degree \(n\) such that the saddle order of the system is at least \(n^2-1\).
scientific article; zbMATH DE number 6370953
scientific article