Cross-ratios and first order ordinary differential equations
Abstract
We investigate the usefulness of the cross-ratio in the study of first order odes of the form y′=Pnyn+Pn−1yn−1+…+P1y+P0, (Pi)0leqslant ileqslant n∈ K, ngeqslant1, where K is a differential field of characteristic zero with algebraically closed field of constants. We discover along the way many differential algebraic identity between solutions to that equation. We also explain why the cross-ratio of any four solutions to such an equation is in general not constant when n≥3 by exhibiting an obstruction which prevents its constancy when n≥3. Finally to any separable polynomial Pφ(z)=zn+1+φ1zn+φ2zn−1+…+φn+1, n≥2, over a differential field of meromorphic function on some domain of ℂ with the usual derivation we associate a uniquely defined first order ode of the form y′=Pnyn+Pn−1yn−1+…+P1y+P0, (Pi)0leqslant ileqslant n∈ K such that every root of Pφ(z) is a solution to the said differential equation.
Related
- Existence and multiplicity results for third-order ordinary differential equations with derivative dependence subject to the Riemann–Stieltjes integral boundary conditions
- Rational linear forms of linearizable ordinary differential equations
- A Kac-Moody root system and linear ordinary differential equations
- The specular ellipse method for scalar ordinary differential equations: exactness and accuracy up to fourth order
- A Sensitivity-based Framework for Calibrating Coupled Ordinary Differential Equations under Model Discrepancy
BibTeX
@misc{wone2026crossratios,
title = {{Cross-ratios and first order ordinary differential equations}},
author = {Oumar Wone},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22712},
eprint = {2610.22712},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22712},
}
RIS
TY - UNPB
TI - Cross-ratios and first order ordinary differential equations
AU - Oumar Wone
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22712
UR - https://arxiv.org/abs/2610.22712
AB - We investigate the usefulness of the cross-ratio in the study of first order odes of the form y^′=P_ny^n+P_n−1y^n−1+…+P_1y+P_0, (P_i)_0leqslant ileqslant n∈ K, ngeqslant1, where K is a differential field of characteristic zero with algebraically closed field of constants. We discover along the way many differential algebraic identity between solutions to that equation. We also explain why the cross-ratio of any four solutions to such an equation is in general not constant when n≥3 by exhibiting an obstruction which prevents its constancy when n≥3. Finally to any separable polynomial P_φ(z)=z^n+1+φ_1z^n+φ_2z^n−1+…+φ_n+1, n≥2, over a differential field of meromorphic function on some domain of ℂ with the usual derivation we associate a uniquely defined first order ode of the form y^′=P_ny^n+P_n−1y^n−1+…+P_1y+P_0, (P_i)_0leqslant ileqslant n∈ K such that every root of P_φ(z) is a solution to the said differential equation.
ER -