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 ℂ and P(f,f(n))=0 for a nonzero polynomial P, then f is rational, elliptic, a rational function of ecz with c≠ 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 DnR(h)=S(h), where D=d/dz and °R=°S≥ 2, then h(z)=Aebz+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 ℚ̅(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 ℂ.
Related
- Entire solutions of higher order Briot-Bouquet equations
- A counterexample to Eremenko's conjecture
- Higher-order jet and its application to the Novikov conjecture
- Analytical Methods for Solving Non-Autonomous Higher-Order Nonlinear Difference Equations
- On higher-order differential equations for Laguerre-Hahn orthogonal polynomials
BibTeX
@misc{marques2026eremenkos,
title = {{Eremenko's conjecture on higher-order Briot--Bouquet equations}},
author = {Diego Marques},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19243},
eprint = {2610.19243},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19243},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Eremenko's conjecture on higher-order Briot--Bouquet equations
AU - Diego Marques
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19243
UR - https://arxiv.org/abs/2610.19243
AB - 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 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≠ 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 °R=°S≥ 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 ℚ̅(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 ℂ.
ER -