PreprintCC BY 4.0arXiv
Rational linear forms of linearizable ordinary differential equations
Abstract
Every scalar ordinary differential equation of order at least three with rational right-hand side that is locally linearizable by a point transformation admits a linear form with rational coefficients over the same coefficient field. We prove this by restricting the derived symmetry algebra to a coordinate line and recovering a scalar differential operator from rational symmetry-jet data. The construction uses differential elimination and linear algebra; it does not require the symmetry generators or a linearizing transformation to be solved for. A single integer parameter suffices to choose the line.
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BibTeX
@misc{lyakhov2026rational,
title = {{Rational linear forms of linearizable ordinary differential equations}},
author = {Dmitry Lyakhov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01338},
eprint = {2610.01338},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01338},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rational linear forms of linearizable ordinary differential equations
AU - Dmitry Lyakhov
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01338
UR - https://arxiv.org/abs/2610.01338
AB - Every scalar ordinary differential equation of order at least three with rational right-hand side that is locally linearizable by a point transformation admits a linear form with rational coefficients over the same coefficient field. We prove this by restricting the derived symmetry algebra to a coordinate line and recovering a scalar differential operator from rational symmetry-jet data. The construction uses differential elimination and linear algebra; it does not require the symmetry generators or a linearizing transformation to be solved for. A single integer parameter suffices to choose the line.
ER -