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The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability; or more precise things like ordinary Galois theory and differential Galois theory.

However, while examples of algebraically closed fields are pretty easy to give (starting with $\mathbb{C}$ and the field of Puiseux series $\varinjlim \mathbb{C}(\!(t^{1/n})\!)$ over it), differentially closed fields... are not. This paper (Spodzieja, “A geometric model of an arbitrary differentially closed field of characteristic zero”) gives an “explicit” construction of one (for some value of “explicit”), but it is highly technical and involves making choices (in a sense in which $\mathbb{C}$ and $\varinjlim \mathbb{C}(\!(t^{1/n})\!)$ do not), and leaves me nonplussed as to why the whole matter needs to be so complicated.

So, with apologies for asking a perhaps vague question: is there some reason why this must be so? Is there some argument why a differentially closed field “must” be complicated to construct?

A little more specifically, while the field of transseries (or any of the many variations thereupon) is “not too far” from being differentially closed, it isn't: is there some reason why any approach to construct a differentially closed variant of transseries must fail? (The answer may be contained in this book, but I failed to locate it.) I could ask the same thing with germs of meromorphic functions at a point.

Gro-Tsen
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    The constuction in the paper is equipping the algebraic closure of some infinite rational function space $\Bbb{Q}(\Lambda_t)$ with a "directional derivative". I believe the index set $T$ can be chosen better to avoid using orderings in the construction, but it is scarcely possible to prove the closedness under differential equations without using choice. – Zerox Mar 03 '22 at 14:01
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    The complexity can be seen from that, in order to construct a nontrivial differential operator, we must introduce some transcendental element $t$ out of nowhere to the usual number field. – Zerox Mar 03 '22 at 14:06
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    Similar complexity caused by '$t$' occurs when completing the value-at-infinity order of real rational functions. – Zerox Mar 03 '22 at 14:07
  • @Zerox Are you claiming, for example, that it is consistent with ZF (no Choice) that $\mathbb{C}(t)$ has no differential closure? Because that (if correct) would certainly answer my question! – Gro-Tsen Mar 03 '22 at 14:55
  • I'm afraid there is no valid answer, because even in the algebraically closed case the equivalence between $AC$ and the existence of such closed field is still open, see this question. Special case do exist like $\Bbb{C} \cong \Bbb{R}[i]$ can be proven to be algebraically closed without $AC$, however topological properties are required. – Zerox Mar 03 '22 at 15:05
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    @Zerox That's not correct. Existence of algebraic closures is known to not be equivalent to AC. It is also known that existence of algebraic closures in general is not provable in ZF alone, though it is provable for a lot of fields one encounters in practice, like subfields of $\mathbb C$ or fields of rational functions over them. – Wojowu Mar 03 '22 at 15:29
  • @Wojowu Thanks for the correction! Do you have any result about the constructibility of differential closed field in $ZF$? According to your statement arbitrary existence is not provable, but how about the particular case of $\Bbb{C}(t)$? – Zerox Mar 03 '22 at 15:40
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    At least the existence of the algebraic closure of $\mathbb{C}(t)$ follows from ZF, since it is the relative algebraic closure of $\mathbb{C}(t)$ in the field of Puiseux series (which is algebraically closed by a theorem of Newton), so no axiom of choice is needed here. This is why it would be interesting to know if the existence of its differential closure follows from ZF. – Gro-Tsen Mar 03 '22 at 18:21
  • I don't think there is anything like that in the transseries book, but maybe it's possible to do some kind of formal power series construction. – Erik Walsberg Mar 04 '22 at 00:20
  • David Marker's introduction to the model theory of differential fields, especially the first and last two pages, may be helpful here for anyone unfamiliar with this subject: http://library.msri.org/books/Book39/files/dcf.pdf –  Mar 04 '22 at 20:41

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