14 GUILLAUME DUVAL Every Picard-Vessiot extension is weakly normal (F Gal∂(F/K) = K), which is a stronger condition than the hypotheses of the theorem. If F/K is a Picard-Vessiot extension, F coincides with the field of fractions of T (F/K). Therefore Case 2 of the theorem cannot occur in the case of a non-trivial valuation of F . Proposition 16(i) was used to prove Theorem 1 but this algebraic statement admits the following analytic interpretation. Let K be the field of meromorphic functions on some connected Riemann sur- face. Let f be an element of K and set w1 = exp exp(f) and w2 = exp exp f. If w1 (respectively w2) is a solution of a monomial linear homogeneous differ- ential equation of order n over K, then u1 = exp(f), (respectively u2 = exp f), is algebraic over K, of degree less than or equal to n. (Take α = f or α = f respectively in Proposition 16(i) and use inequality (9) of the proof.) This prop- erty is closely related to Abel’s problem (see [27] p. 124) for comments and further references. Proposition 16(ii) is a generalisation of a property of Picard-Vessiot Rings in the classical Picard-Vessiot theory. For any Picard-Vessiot extension F/K, let L = 0 be a l.d.e. of order n over K defining F . That is, the solution space V = SolF (L = 0) is a C-vector space of dimension n, and F is generated over K by the elements of V and their derivatives up to the order n 1. In this situation, let (z1,...,zn) be a C-basis of V . Following Singer and Van der Put the Picard-Vessiot Ring associated to L is the following K-algebra R where, w = W (z1,...,zn): R = RL := K[z(j), i 1 i n, 0 j n 1][1/w]. Now, RL is a differential subalgebra of T (F/K), its field of fraction is F and it is simple as a differential algebra (see [27] Chap 1.3 or our proof of Proposition 16 (ii)). Moreover, by ([27] Cor 1.38 p. 30), this last property implies that: (10) RL = T (F/K). In particular R = RL is an intrinsic object of the Picard-Vessiot extension in the sense that it does not depend on the defining l.d.e. L = 0. As a consequence it proves that T (F/K) is a K-algebra of finite type which can be called the Picard- Vessiot Ring of F/K. 2.3. Stability properties for Picard-Vessiot extensions. The main result of this section is Theorem 17. Let F/K be a Picard-Vessiot field extension with an algebraically closed constant field C, and a differential Galois group G(C). i. G(C) has a locally-finite action on S(F/C)∗. Furthermore the following in- equality holds: for all ν S(F/C)∗ and all z T (F/K), (11) card{ν(σ(z))|σ G(C)} ordK(z). ii. Let’s denote by G◦(K) the group of K-points of the connected component of the identity of the C-algebraic group G. Then, there exists a natural action of G◦(K) on T (F/K) (by non differential automorphisms).
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