2. INVARIANT VALUATIONS AND SOLUTIONS OF L.D.E. 13 Therefore, we get the following inductive property (In) for all n : w(n) = Pn(u) · w, where Pn(u) is a monic polynomial expression of degree n in u with coefficients depending on α and its derivatives. Since w T (F/K), it satisfies a homogeneous linear differential equation with coefficients in K: w(n) + an−1 · w(n−1) + · · · + a1 · w + a0 · w = 0. Hence, Pn(u) · w + an−1 · Pn−1(u) · w + · · · + a1 · u · w + a0 · w = 0 So, (8) Pn(u) + an−1 · Pn−1(u) + · · · + a1 · u + a0 = 0. Equation (8) is a monomial algebraic relation of degree n in u, with coefficients depending on the ai’s, α, and their derivatives. Since α is algebraic over K, K[α]/K is a differential field extension. Therefore, all the coefficients of (8) belong to K[α]. Hence, u is algebraic over K, and we have the inequality (9) [K[u]/K] [K[α]/K] · ordK(w) ∞. Hence, w w = u is algebraic over K and w is a unit of T (F/K) by Harris-Sibuya’s theorem (see Proposition 15 (iv)). (ii) Let I be a nonzero ideal of T (F/K), and let z = 0 belong to I. Let V (z) be the C-vector space spanned by the conjugates of z under Gal∂(F/K). Then, V (z) is a finite dimensional vector space by Proposition 15(ii). Let {z = z1,...,zd} be some of the conjugates of z which form a C-basis of V (z). Let w = W (z1,...,zd) be the associated Wronskian determinant as in (7). Now assume that I is a differential ideal, (respectively a Gal∂(F/K)-module). By expanding w by the first column, (respectively by the first row), one gets in both cases that w I. To conclude the proof of (ii), we just have to prove that the condition F Gal∂(F/K) T (F/K) implies that w is a unit of T (F/K). The linear representation ρ : Gal∂(F/K) −→ GlC(V (z)) gives: σ(w) = det ρ(σ) · w σ(w ) = det ρ(σ) · w σ( w w ) = w w Hence, u = w w F Gal∂(F/K), which is a field. So if F Gal∂(F/K) T (F/K), u is a unit of T (F/K), and so is w by (i) and Proposition 15(i). Proof of Theorem 1 and Corollary 2. Let ν be a non-trivial G-invariant valuation of F . If T (F/K) is not contained in Rν, by definition there exists some t in T (F/K)∗ such that ν(t) 0, and Case 1 of Theorem 1 holds. Now suppose that T (F/K) Rν. Since ν is G-invariant, is a G-stable ideal of Rν. The center P = T (F/K) of ν in T (F/K), is therefore a prime G-stable ideal of T (F/K). By Proposition 16(ii), it follows that P is trivial i.e. P = 0 or P = T (F/K). Since ν is non-trivial, 1 P , and we can exclude the case P = T (F/K). So we have P = T (F/K) = {0}. Which gives T (F/K)∗ U(Rν) and proves the theorem. The proof of the corollary follows from the two statements below:
Previous Page Next Page