Discovering Polynomial and Quadratic Structure in Nonlinear Ordinary Differential Equations

The paper defines polynomialization and differentially-algebraic functions, and states Theorem 1: if f is differentially-algebraic with respect to x on D, then a polynomialization exists, citing prior work for the proof. The chapter does not provide a full proof, only the statement and context (Definition 1 and 2, Theorem 1) with examples and broader discussion . The candidate solution supplies an independent proof sketch via differential algebra: build a (radical) differential ideal from the single-variable differential equations, take a characteristic set, use Ritt reduction to represent higher derivatives via finitely many “basic jets” after multiplying by powers of the product of initials and separants S, and then close the Lie derivative D = L_f on a finite list w by introducing weighted jets S^M·(δ^α f). This yields polynomial right-hand sides for ẋ and ẇ without divisions, hence a polynomialization. The approach aligns with the paper’s assumptions and conclusion but uses a different technique than the chapter (which defers to the literature), so both are correct with different proofs.