The Thue-Siegel-Roth-Schmidt theorem for algebraic functions

Ratliff, Michael I.

doi:10.1016/0022-314x(78)90012-4

Cited by 11 publications

(4 citation statements)

References 2 publications

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“…For instance, for any field K of zero characteristic the continued fraction C(x), viewed as an element of K((1/x)), is transcendental over K(x). This is a consequence of the function field analogue in zero characteristic of Roth's theorem obtained in [11]. Notice also that the continued fraction C(q) converges for every complex number q with |q| > 1.…”

Section: Introductionmentioning

confidence: 66%

Non-converging continued fractions related to the Stern diatomic sequence

Adamczewski¹

2010

Acta Arith.

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Section: Introductionmentioning

confidence: 66%

Non-converging continued fractions related to the Stern diatomic sequence

Adamczewski¹

2010

Acta Arith.

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“…We refer to the appropriate reformulation of Schmidt's result in the functional case as "Schmidt's theorem." Such ineffective Schmidt's theorem was proved by Ratliff (6).…”

Section: Q(x)mentioning

confidence: 93%

Rational approximations to solutions of linear differential equations

Chudnovsky¹,

Chudnovsky²

1983

Proc. Natl. Acad. Sci. U.S.A.

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Rational approximations of Pade and Pade type to solutions of differential equations are considered. One of the main results is a theorem stating that a simultaneous approximation to arbitrary solutions of linear differential equations over C(x) cannot be "better" than trivial ones implied by the Dirichlet box principle. This constitutes, in particular, the solution in the linear case of Kolchin's problem that the "Roth's theorem" holds for arbitrary solutions of algebraic differential equations. Complete effective proofs for several valuations are presented based on the Wronskian methods and graded subrings of Picard-Vessiot extensions.The purpose of this paper is to present simple and complete proofs.of several results on best rational approximations to solutions of differential equations. Our methods are based on studies of particular differential invariants associated with graded subrings of Picard-Vessiot extensions of differential fields (1). The origin of these studies is the famous Kolchin's problem (2) on the extension of "Roth's theorem" to approximation of solutions of arbitrary algebraic differential equations by rational functions. The Roth's theorem is used here in the broad sense as the property of numbers or functions that was proved by Roth (3) for algebraic numbers. For numbers 6, the Roth's theorem means that for any E > 0 and arbitrary rational integers p and q, 10 -(p/q)l > IqL 2-8 for q 2 qo(E). For functions y(x), the Roth's theorem means that for any E > 0 and arbitrary poly-P(x) nomials P(x) and In this paper, we prove functional solutions of linear differential equations over k(x) and also prove Schmidt's theorem and the generalization of Roth's theorem for the simultaneous approximations in several normings. The methods of this paper are also useful in our complete solution of the Kolehin problem for arbitrary algebraic differential equations. We want to acknowledge that C. Osgood announced a year ago an effectivization of Roth's theorem for algebraic functions and recently announced the Roth's and Schmidt's theorems for solutions of linear differential equations. After the Roth theorem (3) in 1955, its p-adic and g-adic generalizations appeared (11)(12)(13). This development was summarized by Mahler in his book (11), where he presented a general "approximation theorem" on rational approximations of algebraic numbers in archimedian and nonarchimedian metrics.Let Pi, P2, . Pr+r'+ r be a fixed system of r + r' + r" distinct primes. Also, for a real number and for a p-adic number

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“…v(a,/3) = 1 for every pair (a,/3) of algebraic elements of K((T-1)) such that 1, a,/3, are linearly independent over K(T) ( [1], [4]). …”

Section: Let A~k((t-1))mentioning

confidence: 99%