Throughout the paper all fields are assumed to be of characteristic zero and "differential" means "ordinary differential". Standard proofs of the primitive element theorem [1, Ⅴ, Theorem 4.6] and the Noether normalization lemma [1, Ⅷ, Theorem 2.1] are based on a consideration of "generic combinations" of initial generators. We propose a differential counterpart of this argument which we call a "polynomial shifting" trick. It is an important part of recent proofs of a strengthened version of Kolchin's primitive element theorem (see [3, Theorems 1 and 2]) and a differential analog of the Noether normalization lemma (see [4, Theorem 1]). This trick turned out to be quite flexible and constructive. We hope that this method will be useful dealing with problems of the same flavour.
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