Algebraic Geometry and Differential Equations

[gen. granger]

I'm doing some work on Differential Galois Theory and was wondering whether the following observation is valid or not.

For a Linear Scale Differential Equation (LSDE) over a differential field F, equipped with derivation ', of the form
L(y) := y^(n) + a_(n-1)y^(n-1) + a_1y' + a_0 = 0

we have a solution space V := {v in F : L(v) = 0}, i.e. the list of elements in the field F, or in some differential extension of F, such that the equation holds true.

Would it be fair to assume that this solution space is an affine variety? If this is the case, then would the normal apparatus used in Algebraic Geometry be used in order to find the torsor for this equation and thus in turn the Differential Galois Group that would be associated with this equation?

I think this a correct assumption to have. If it is not, I would really appreciate some guidance on the matter as the material I have on the subject is far from descriptive in recovering the Galois groups of Differential Equations when it comes to examples.

Thanks in advance

Gen. Granger

 

[pat_connell]

Yes, the solution space V is indeed an affine variety. Therefore, you can use the normal apparatus of Algebraic Geometry to find the torsor for this equation and thus in turn the Differential Galois Group that would be associated with this equation. You can find some helpful resources on differential Galois theory and related topics in books such as "Differential Galois Theory" by Vladimir V. Kisil.