AI 中文总结
本文证明所有至少三阶且右端有理、局部可点变换线性化的标量常微分方程，在同一系数域上存在有理系数的线性形式，方法基于对称代数限制与微分消元。
AI 中文摘要
每个至少三阶的标量常微分方程，若其右端为有理函数且在局部上可通过点变换线性化，则该方程在同一系数域上存在一个具有有理系数的线性形式。我们通过将导出的对称代数限制在一条坐标线上，并从有理对称-射流数据中恢复一个标量微分算子来证明这一点。该构造使用微分消元和线性代数；它不需要求解对称生成元或线性化变换。一个整数参数即可用于选择该线。
英文摘要
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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