On nonexistence of algebraic curve solutions to second-order polynomial autonomous systems over the complex field.
Consider the polynomial system \[ \frac{dx}{dt}=y,\quad \frac{dy} {dt}=y \sum^\ell_{i=1}c_ix^i+\sum^\ell_{i=1}g_ix^i\tag{*} \] with \(c_i,g_i\in\mathbb{C}\), \(c_\ell g_\ell\neq 0\). The author proves that (*) has no solution of the form \[ \sum^m_{i=0}a_i(x)y^i=0, \] where \(a_i\) are polynomials in \(x\).
scientific article; zbMATH DE number 1921204