J Austral Math Soc Ser A 50 pp320--332, 1991.

A Subspace Theorem for Ordinary Linear Differential Equations

Alice Ann Miller

Abstract

The study of the S-unit equation for algebraic numbers rests very heavily on Schmidt's Subspace Theorem. Here we prove an effective subspace theorem for the differential function field case, which should be valuable in the proof of results concerning the S-unit equation for function fields. Theorem 1 states that either

Orda(L1L2...Ln)

has a given upper bound where

L1(P), L2(P), ..., Ln(P)

are linearly independent linear forms in the polynomials

P = (P1(x), P2(x), ..., Pn(x))

with coefficients that are formal power series solutions about x = 0 of a non-zero differential equations and where Ord_a denotes the order of vanishing about a regular (finite) point of functions f_k,i : (k = 1, n ; i = 1, n) or

P = (P1(x), P2(x), ..., Pn(x))

lies inside one of a finite number of proper subspaces of (K(n))ⁿ. The proof of the theorem is based on the wroskian methods and graded sub-rings of Picard-Vessiot extensions developed by D. V. Chudnovsky and G. V. Chudnovsky in their function field analogues of the Roth and Schmidt theorems. A brief discussion concerning the possibility of a subspace theorem for a product of valuations including the infinite one is also included.

1980 AMS Subject Classification (1985 Revision): primary 11J61, 11J82, 11Q10

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Authors

Alice Ann Miller

University of Western Australia, Nedlands WA 6009, Australia.