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Do the relations between Galois groups and solutions to polynomial equations with one variable have a counterpart between Lie groups and solutions to differential equations ?

Do the relations between algebraic variaties and solutions to polynomial equations have a counterpart between differential manifolds and solutions to differential equations ?

Is there a similar theorem/counterpart like Matiyasevich’s theorem for differential equations ?

Any other parallel theorem, idea, theory exist for differential equations as those for polynomial equations ?

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    $\begingroup$ Sure, there is a whole area of differential Galois theory, in which the Galois groups are linear algebraic groups. The category of finite extensions of a field is replaced with that of finite differential modules. In characteristic zero, this is a Tannakian category and thus is is the category of representations of a proalgebraic group. $\endgroup$
    – Piotr Achinger
    Commented Nov 14 at 7:37
  • $\begingroup$ I wonder what the counterpart of differential equations has to be to the scheme of polynomials $\endgroup$
    – XL _At_Here_There
    Commented Nov 14 at 7:53

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To (Q1): Yes, see Differential Galois Theory. For example, the Galois group of the Airy equation over $\mathbb{C}$ is given by the Lie group $\operatorname{SL}_2(\mathbb{C})$.

To (Q2): Yes, see (Q4).

To (Q3): No idea. What would be a Diophantine differential set?

To (Q4): Yes, see Vinogradov's Diffieties. This kind of analogy can be taken very far, see for example the recent fascinating work of Kryczka, Sheshmani, and Yau:

Derived Moduli Spaces of Nonlinear PDEs I: Singular Propagations

Derived Moduli Spaces of Nonlinear PDEs II: Variational Tricomplex and BV Formalism

The references therein are also worth checking out.

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  • $\begingroup$ Possibly they are polynomials or rational function over $\mathbb{C}$ $\endgroup$
    – XL _At_Here_There
    Commented Nov 14 at 7:47
  • $\begingroup$ Differential manifolds are like variaties? $\endgroup$
    – XL _At_Here_There
    Commented Nov 14 at 7:49
  • $\begingroup$ @XL_At_Here_There: to your first comment, possibly yes, but I wouldn't know how to state an analogue of Matiyasevich’s theorem. To your second comment: again, it depends on what you mean by this :-) For example, by the Nash-Tognoli theorem every closed smooth manifold is diffeomorphic to a real algebraic variety. It seems to me that the concept of deffieties is the analogy you are looking for. But in this context the analogue of varieties is not differential manifolds, but rather infinite-dimensional differential manifolds. (cont.) $\endgroup$
    – M.G.
    Commented Nov 14 at 8:21
  • $\begingroup$ @XL_At_Here_There: (cont.) The points on differential manifolds are not the solutions of a PDE, but rather differential manifolds supply you with the functions which are acted upon by differential operators and the latter are subsets of the jet bundles of the manifold. That is, you have an additional level compared to varieties: points -> functions -> differential operators. $\endgroup$
    – M.G.
    Commented Nov 14 at 8:23
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The answer to Q2 is provided by Differential algebra created by Joseph Fels Ritt. He studied differential-algebraic varieties by analogy with algebraic ones.

See Ritt, Joseph Fels, Differential algebra. Dover Publications, Inc., New York, 1966, and

Kolchin, E. R. Differential algebra and algebraic groups. Academic Press, New York-London, 1973.

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  • $\begingroup$ Thanks for your answer and nice to meet you again. $\endgroup$
    – XL _At_Here_There
    Commented Nov 14 at 15:24

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