Simultaneous Approximation by Conjugate Algebraic Numbers in Fields of Transcendence Degree One
نویسنده
چکیده
We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers ξ1, ..., ξt by conjugate algebraic numbers of bounded degree over Q, provided that the given transcendental numbers ξ1, ..., ξt generate over Q a field of transcendence degree one. We provide sharper estimates for example when ξ1, ..., ξt form an arithmetic progression with non-zero algebraic difference, or a geometric progression with non-zero algebraic ratio different from a root of unity. In this case, we also obtain by duality a version of Gel’fond’s transcendence criterion expressed in terms of polynomials of bounded degree taking small values at ξ1, ..., ξt.
منابع مشابه
Diophantine approximation by conjugate algebraic integers
Building on the work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel’fond’s transcendence criterion which provides a sufficient condition for a complex or p-adic number ξ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at ξ together with most of their derivatives. The second one, which follows from this crit...
متن کاملSimultaneous Approximation of Logarithms of Algebraic Numbers
Recently, close connections have been established between simultaneous diophantine approximation and algebraic independence. A survey of this topic is given by M. Laurent in these proceedings [7]. These connections are one of the main motivations to investigate systematically the question of algebraic approximation to transcendental numbers. In view of the applications to algebraic independence...
متن کاملSimultaneous Approximation and Algebraic Independence
We establish several new measures of simultaneous algebraic approximations for families of complex numbers (θ1, . . . , θn) related to the classical exponential and elliptic functions. These measures are completely explicit in terms of the degree and height of the algebraic approximations. In some instances, they imply that the fieldQ(θ1, . . . , θn) has transcendance degree≥2 overQ. This appro...
متن کاملCriteria for irrationality, linear independence, transcendence and algebraic independence
For proving linear independence of real numbers, Hermite [6] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [14] is dual (duality in the sense of convex bodies): he considers simultaneous approximation by means of independent linear forms. We define the height of a linear form L = a0X0 + · · · + amXm with complex coeff...
متن کاملDiophantine approximation , irrationality and transcendence Michel Waldschmidt
For proving linear independence of real numbers, Hermite [18] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [34] is dual (duality in the sense of convex bodies): he considers simultaneous approximation by means of independent linear forms. We define the height of a linear form L = a0X0 + · · · + amXm with complex coef...
متن کامل