Source-linked AI summary
Rational linear forms of linearizable ordinary differential equations
Dmitry Lyakhov
TL;DR
The paper asks whether local point-linearizability of a rational scalar ODE guarantees a linear form over its original coefficient field. It recovers an operator from symmetry-jet data and specializes it to a coordinate line chosen by an integer. The result is a rational linear form, without requiring the linearizing transformation itself to be rational.
Problem
The paper addresses whether a locally point-linearizable scalar ODE with rational right-hand side has a linear form over the same coefficient field.
Source: §1 · p. 1Method
The proof uses symmetry-jet data to recover a scalar operator and specializes its coefficients to a coordinate line selected by an integer.
Source: §2 · pp. 3–4Results
Every such equation has a linear form with rational coefficients over the same coefficient field.
Source: §1 · pp. 1–2Takeaways & Limitations
The theorem guarantees a rational linear form, not rationality of every linear form or of the linearizing transformation.
Source: §1 · p. 2The result applies to equations locally linearizable by a locally invertible analytic point transformation and guarantees existence, not rationality of the map or every linear form.
Source: §1 · pp. 1–2
Abstract
from arXiv · showhide
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.
1 Statement and background
For scalar rational ODEs of order at least three, local point-linearizability guarantees a linear form over the same coefficient field. The paper uses symmetry structure to recover that form and distinguishes its rationality from that of the linearizing transformation.
1 Statement and background
- The equation is considered linearizable when a locally invertible analytic point transformation maps it to a linear equation.Equivalences are local, away from singularities and where the prolonged transformation is defined.Source: §1 · p. 1
- Every locally linearizable scalar ODE of order n ≥ 3 with rational right-hand side has a linear form with rational coefficients over the same coefficient field.The theorem guarantees existence of such a form, not rationality of every linear form or of the linearizing transformation.Source: §1 · pp. 1–2
- The form can be computed by differential elimination and linear algebra without solving differential equations or constructing a linearizing transformation.The algorithm requires effective exact arithmetic and zero tests in the coefficient field.Source: §1 · p. 2
- For n ≥ 3, the stated symmetry criterion is m = n + 4, or m ∈ {n + 1, n + 2} with an n-dimensional abelian derived algebra.Here m is the dimension of the Lie point symmetry algebra.Source: §1 · p. 2
- Abstract symmetry algebra alone cannot determine variable coefficients in the m = n + 1 case, so the proof uses its realization by vector fields and rational jet equations.The m = n + 2 constant-coefficient recovery case was studied in [3].Source: §1 · p. 2
2 Proof of the theorem
The proof encodes symmetries through rational jet data, extracts an operator annihilating the derived algebra, and specializes its coefficients to a coordinate line. It then verifies locally that the resulting scalar equation is linear and equivalent.
2 Proof of the theorem
- Involutive reduction of the symmetry determining system produces rational jet data and a compatible first-order system whose solutions are determined by arbitrary initial vectors.The data include matrices for the coordinate derivatives and a bilinear map encoding Lie brackets.Source: §2 · pp. 2–3
- Outside the maximal case, the derived algebra has dimension n, and a basis is represented by the columns of E.The maximal case returns w^(n) = 0.Source: §2 · p. 3
- A horizontal or vertical line is selected according to the symmetry component, so the restricted basis has a nonzero Wronskian and the matrix M is invertible.The Wronskian argument uses the local linearizing coordinates; invertibility of M then follows from W = det M det C.Source: §2 · pp. 3–4
- The operator coefficients are recovered by b = −R_n E M^−1 from the ambient jet derivatives, without differentiating the moving coordinates.The resulting relation annihilates the derived-algebra symmetry data.Source: §2 · p. 4
- Specialization to a horizontal or vertical line yields coefficients in K(λ), and some integer choice of the fixed coordinate avoids the finitely many excluded conditions.Only finitely many integer values are excluded by denominators, regularity factors, rank, or determinant conditions.Source: §2 · p. 4
- Locally straightening the commuting rank-one symmetry distribution makes the transformed equation linear; subtracting a particular solution removes any inhomogeneous term.The restricted symmetry components have a nonzero Wronskian and solve the recovered operator equation.Source: §2 · pp. 4–5
3 Computation and an example
The paper gives a finite procedure for testing linearizability and recovering the rational form, then illustrates the recovery in an example. Its example retains rational coefficients even though the displayed linearizing transformation is transcendental.
3 Computation and an example
- The procedure completes the determining system, checks the symmetry-algebra conditions, computes b, and tries integer specializations until the required conditions hold.A failed algebraic test returns False; a successful specialization returns the linear form with specialized coefficients.Source: §3 · p. 5
- The procedure terminates under the effective-field hypothesis because involutive completion and matrix computations are finite and only finitely many integers are excluded.The search needs no a priori degree bound; specializing the jet data first is offered as an efficiency option.Source: §3 · p. 5
- In the example, the cyclic component η gives b = (−x, 0, 0), and the recovered coefficients remain over Q despite the transcendental linearizing transformation.The output is obtained by specializing on y = c.Source: §3 · p. 6