On realizing differential-algebraic equations by rational dynamical systems

The paper’s main theoretical result (Theorem 4.2) states that if an irreducible input–output differential polynomial % of order h has a rational (resp., input‑affine rational) realization, then it admits one of dimension h; the proof flows through an equivalence with the existence of a dominant map to the hypersurface H% together with solvability of a Lie-derivative-based linear system (Lemma 4.1), and then extracts h algebraically independent coefficient functions via Lemma 4.3/Corollary 4.4 to build an h-state realization, with the input‑affine case handled by a block-linear system and Cramer’s rule (Proposition 4.5/4.6) . The candidate solution proves the same statement by (i) giving a lower bound n ≥ h via differential transcendence degree and (ii) constructing an h‑state realization using precisely the same Lie-derivative criterion and coefficient-independence/Jacobian-rank argument as in the paper. The lower bound n ≥ h is implicit in the paper as well (any realization yields a dominant map A^n → H%, so n ≥ dim H% = h by Lemma 4.1), though the candidate spells it out using Kolchin–Ritt dimension theory. Overall, both are correct and follow substantially the same proof strategy; the model adds a standard lower‑bound argument the paper leaves implicit.