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A subspace theorem for ordinary linear differential equations

  • Alice Ann Miller (a1)

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 has a given upper bound where are linearly independent linear forms in the polynomials with coefficients that are formal power series solutions about x = 0 of non-zero differential equations and where Orda denotes the order of vanishing about a regular (finite) point of functions ƒk, i: (k = 1, n; i = 1, n) or lies inside one of a finite number of proper subspaces of (K(x))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.

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References

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[1]Chudnovsky, D. V. and Chudnovsky, G. V., ‘Rational approximations to solutions of linear differential equations’, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), 51585162.
[2]Evertse, J. H. and Györy, K., ‘S-unit equations and their applications’, New advances in transcendence theory (Durham, 1986), pp. 110174, (Cambridge University Press, Cambridge, New York, 1988).
[3]Jacobson, N., Basic algebra II, (Freeman, 1980).
[4]Osgood, C. F., ‘Concerning a possible Thue-Siegel-Roth theorem for algebraic differential equations’, Number theory and algebra, pp. 223234, (Academic Press, 1977).
[5]Osgood, C. F., ‘Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better’, J. Number Theory 21 (1985), 347389.
[6]Roth, K. F., ‘Rational approximations to algebraic numbers’, Mathematika 2 (1955), 120.
[7]Schlickerwei, H. P., ‘On products of special linear forms with algebraic coefficients’, Acta Arith. 31 (1976), 389398.
[8]Schmidt, W. M., ‘Diophantine approximations’, Lecture Notes in Mathematics 785, (Springer-Verlag, 1980).
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A subspace theorem for ordinary linear differential equations

  • Alice Ann Miller (a1)

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