One advance in the proof of the conjecture on meromorphic solutions of Briot–Bouquet type equations

We study entire solutions (solutions which are entire functions) of differential equations of the form $P(y,y^{(n)})=0$, where $P$ is a polynomial with complex coefficients, $n$ is a natural number. We show that, under some constraints on $P$, all entire solutions of such equations are either polynomials, or functions of the form $e^{-L\beta z}Q(e^{\beta z})$, where $L$ is a nonnegative integer, $\beta$ is a complex number, and $Q$ is a polynomial with complex coefficients. This verifies the well-known A. E. Eremenko's conjecture on meromorphic solutions of autonomous Briot–Bouquet type equations for entire solutions in the nondegenerate case.