Constrained extensions of real type

Picard-Vessiot Extensions of Real Differential Fields

“…K ⊂ E ⊂ L, then L|E is a real Picard-Vessiot extension and DGal(L|E) is a C-defined closed subgroup of DGal(L|K). As in the ordinary case (see [5] Theorem 3.1, [6] Theorem 4.4), we obtain a Galois correspondence theorem. Theorem 4.…”

“…, u m . Now, by [5] Corollary 2.6, there exists a real field F which is a Picard-Vessiot extension of K D for (2). Clearly F (i) is a Picard-Vessiot extension of K D (i) for (2).…”

“…In [5] and [6], we proved the existence of a real Picard-Vessiot extension for linear differential equations defined over a real ordinary differential field with a real closed field of constants C, we gave an appropriate definition of its differential Galois group, proved that it has the structure of a C-defined linear algebraic group and established a Galois correspondence theorem in this setting. In [4], we gave a characterization of real Liouville extensions of an ordinary differential field in terms of differential Galois groups, which answers a question raised in [8].…”

Real Liouville Extensions

“…In [4] and [5], we proved the existence of a Picard-Vessiot extension for linear differential equations defined over a real differential field with a real closed field of constants C, we gave an appropriate definition of its differential Galois group, proved that it has the structure of a C-defined linear algebraic group and established a Galois correspondence theorem in this setting. In [6], Gel'fond and Khovanskii characterized Liouville functions over R using differential rings of real functions with a finiteness property.…”

The unicity of real Picard-Vessiot fields

“…A real Picard-Vessiot field L for M/K is a Picard-Vessiot field which is also a real field. In [CHS1] and [CHS2] the existence of a real Picard-Vessiot field is proved using results of Kolchin.…”