Differential Weil descent

Differentially large fields

“…For differentially large fields, this can also be carried out. We will explain a special case of the differential Weil descent suitable for our purpose and refer to [19,Theorem 3.4] for the general assertion and for proofs.…”

“…Here homomorphisms are algebra homomorphisms over K and L, respectively. Now in [19,Theorem 3.4] it is shown that the ring W (S) can be naturally expanded to a differential K -algebra W diff (S) such that the bijection above restricts to a bijection…”

“…However for our application below only the existence of the bijections above are needed. We refer to [19,Section 3] for a self contained exposition of the matter, where all data are constructed explicitly. In particular the construction there shows that W diff (S) is a differentially finitely generated K -algebra if S is a differentially finitely generated L-algebra.…”

“…We lay out the foundations of this new and exciting class of "tame" differential fields, prove several characterizations (see Theorem 4.3, Proposition 4.7), and exhibit many examples (see Proposition 4.7, Corollary 4.8(ii), Theorems 5.12 and 5.18, and 5.2) and applications (see Corollaries 4.8(iii) and 5.13, Theorems 5.7 and 5.12, Lemma 5.9, Proposition 5.16, and 5.14 and 5. 19). An outline of these is given in the rest of the introduction.…”