Rational Approximations to Algebraic Functions

Uchiyama, Saburô

doi:10.14492/hokmj/1530756192

Cited by 21 publications

(16 citation statements)

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“…Now let K be the field of formal power series with coefficients in a field k of characteristic zero, and let l a 1 be the non-archimedean absolute value with 101 = 0 and with la( = e p if a,# 0. All the results stated above remain true (see [9]) if integers p, q are replaced by polynomials (i.e., elements of the ring k [ t ] ) , if rationals are replaced by rational functions (i.e., elements of the field k ( t ) ) , real algebraic numbers are replaced by algebraic functions (i.e., quantities which are algebraic over k( t ) ) in K , and if the ordinary absolute value is replaced by our non-archimedean absolute value. It was first noted by Kolchin [3] and by Osgood [ 5 ] , [6] that approximation results can be proved for K which either have no analogue for the real numbers, or for which such an analogue is beyond the scope of present methods.…”

Section: Introductionmentioning

confidence: 60%

On Osgood's effective thue theorem for algebraic functions

Schmidt

1976

Comm Pure Appl Math

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Section: Introductionmentioning

confidence: 60%

On Osgood's effective thue theorem for algebraic functions

Schmidt

1976

Comm Pure Appl Math

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“…By choosing one of the homogeneous coordinates equal to 1, one can show that (14) h(x);:;:::: 1 for xeP 2 (K) and (15) h(x) = 1 if and only if xeP 2 (k).…”

Section: Remains True Withmentioning

confidence: 99%

“…[7], Ch. 7, [5], [15]). In this paper we shall give analogues of the results in [2] for algebraic function fields of characteristic 0.…”

Section: Introductionmentioning

confidence: 99%

On equations in two S-units over function fields of characteristic 0

Evertse¹

1986

Acta Arith.

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“…The Roth's theorem for algebraic functions, as ineffective as for algebraic numbers, was proved by Uchiyama (4). Later, Schmidt generalized the Roth theorem for the case of simultaneous approximations to several algebraic numbers (5).…”

Section: Q(x)mentioning

confidence: 99%

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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Rational Approximations to Algebraic Functions

Cited by 21 publications

References 0 publications

On Osgood's effective thue theorem for algebraic functions

On Osgood's effective thue theorem for algebraic functions

On equations in two S-units over function fields of characteristic 0

Rational approximations to solutions of linear differential equations

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