Eremenko's conjecture on higher-order Briot--Bouquet equations

Abstract

The Briot--Bouquet theorem describes the meromorphic solutions of autonomous first-order algebraic differential equations. We prove Eremenko's conjecture extending it to all orders: if $f$ is meromorphic in $\mathbb{C}$ and $P(f,f^{(n)})=0$ for a nonzero polynomial $P$, then $f$ is rational, elliptic, a rational function of $e^{cz}$ with $c\neq 0$, or satisfies $f^{(n)}=af+b$. In particular, every meromorphic solution with $P$ irreducible of degree at least two belongs to the Weierstrass class. Solutions with poles were classified by Eremenko, Liao and Ng. We settle the remaining entire case with a rigidity theorem for polynomial compositions: if $h$ is nonconstant entire and $D^nR(h)=S(h)$, where $D=d/dz$ and $\deg R=\deg S\ge 2$, then $h(z)=Ae^{bz}+Be^{-bz}+C$. No growth assumption is needed. The proof turns the cancellation of leading terms into a contraction of indicator diagrams and obtains incompatible zero counts near an exposed edge. As an arithmetic application, an algebraic Laurent expansion at one algebraic point and an algebraic value at another force a solution to lie in $\overline{\mathbb{Q}}(z)$. Hence any two distinct derivatives of a function in Mahler's class (transcendental meromorphic functions with algebraic Laurent coefficients at every algebraic point) are algebraically independent over $\mathbb{C}$.