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:
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