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Picard-Vessiot extensions of real differential fields. (English) Zbl 1447.12004
Let $$K$$ be an ordinary differential field of characteristic zero with a field of constants $$k$$, $$P$$ be a linear differential operator over $$K$$ and $$\eta_{1},\ldots,\eta_{n}$$ is a fundamental system of zeros of $$P$$. The field $$L=K\langle\eta_{1},\ldots,\eta_{n}\rangle$$ is called a Picard-Vessiot extension of the field $$K$$, if the constants $$L$$ and $$K$$ coincide. When $$k$$ is algebraically closed, according to the differential Galois theory, such an extension for $$P$$ always exists and it is unique up to $$K$$-differential isomorphism. In this case, the operator $$P$$ can be assigned a group G of $$K$$-differential automorphisms by setting $$G=\mathrm{DGal}(L/K)$$ which has the structure of a linear algebraic group defined over $$k$$. If the field $$k$$ is not algebraically closed, then the situation is substantially complicated. For example, uniqueness may be violated. In the paper, the authors investigate a such case when $$K$$ is formally real differential field with real closed field of constants $$k$$. In this case for any given connected semi-simple linear algebraic group $$G$$ defined over $$k$$, there exists (see [T. Crespo et al., Math. Ann. 365, No. 1–2, 93–103 (2016; Zbl 1344.34096), 1, Proposition 3.3]) a linear differential operator $$P$$ over $$K$$ and a formally real Picard-Vessiot extension $$L/K$$ for it such that $$G=\mathrm{DGal}(L/K)$$. The paper discusses in detail the case when a complexification $$H(H:=G\times_{k}\bar{k})$$ of $$G$$ is one of the classical groups $$\mathrm{SL}$$, $$\mathrm{SU}$$, $$\mathrm{SO}$$ or $$\mathrm{Sp}$$. “The inspection of the different cases shows that there is no general pattern.”
##### MSC:
 12H05 Differential algebra 13B05 Galois theory and commutative ring extensions 14P05 Real algebraic sets 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
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