Real Liouville Extensions

Abstract

We give a characterization of real Liouville extensions by differential Galois groups.

Picard-Vessiot Extensions of Real Differential Fields

“…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 Liouvillian Extensions of Partial Differential Fields

“…Let now K be a formally real field with real closed field of constants C K . From Proposition 3.7 and the corresponding results in the ordinary case [7,Examples 7 and 8], we obtain that, if α is an integral over K, then K⟨α⟩/K is a real Picard-Vessiot extension and its differential Galois group DGal(K⟨α⟩|K) is isomorphic to the additive group G a ; and, if α is the exponential of an integral and the field K⟨α⟩ is real and with field of constants equal to C K , then K⟨α⟩/K is a Picard-Vessiot extension and DGal(K⟨α⟩|K) is isomorphic to the multiplicative group G m , or a finite subgroup of it. Definition 3.12.…”

“…In this section we present the relationship between the concept of real Liouville function and the results on Liouvillian extensions of formally real differential fields treated in [7] and Section 3.3 of this paper. The solutions of differential equations which until now were abstract elements are considered in this section as real functions in real variables and we study their properties as such functions.…”

“…Picard-Vessiot theory for formally real differential fields with real closed field of constants allows then to characterise some aspects of the behaviour of real functions. In [7] T. Crespo and Z. Hajto characterised real Liouvillian extensions of ordinary differential fields in terms of real Picard-Vessiot theory. Their result answers an earlier question of A. Khovanskii, namely "Is it true that a necessary and sufficient condition for solvability of a real differential equation by real Liouville functions follows from real Picard-Vessiot theory?…”

“…However the differential Galois group G for (2.1) over K gives more information than G on the behaviour of the solutions to (2.1). In the case when K is a formally real ordinary differential field and C K is real closed, we obtained in [7] that the property for a system Y ′ = AY defined over K to have solutions which are real Liouville functions is characterised by the differential Galois group G. Therefore it is interesting to extend this result to partial differential fields.…”