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Preprint Number 1689

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1689. Omar León Sánchez and Anand Pillay
Differential Galois cohomology and parameterized Picard-Vessiot extensions
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Abstract:

Assuming that the differential field (K,δ) is differentially large, in the sense of Leon Sanchez and Tressl, and “bounded” as a field, we prove that for any linear differential algebraic group G over K, the differential Galois (or constrained) cohomology set H^1_δ(K,G) is finite. This applies, among other things, to closed ordered differential fields K, in the sense of Singer. As an application, we prove a general existence result for parameterized Picard-Vessiot extensions within certain families of fields; if (K,δ_x,δ_t) is a field with two commuting derivations, and δ_x Z = AZ is a parameterized linear differential equation over K, and (K^{δ_x},δ_t) is “differentially large” and K^{δ_x} is bounded, and (K^{δ_x}, δ_t) is existentially closed in (K,δ_t), then there is a PPV extension (L,δ_x,δ_t) of K for the equation such that (K^{δ_x,δ_t) is existentially closed in (L, δ_t). For instance, it follows that if the δ_x-constants of a formally real differential field (K,δ_x,δ_t) is a closed ordered δ_t-field, then for any homogeneous linear δ_x-equation over K there exists a PPV extension that is formally real.

Mathematics Subject Classification: 03C60, 12H05

Keywords and phrases: differential Galois cohomology, D-varieties, PPV extensions

Full text arXiv 1911.06165: pdf, ps.


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