Baker’s Theorem

Saradha Natarajan () and Ravindranathan Thangadurai
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Saradha Natarajan: University of Mumbai, DAE Centre for Excellence in Basic Sciences
Ravindranathan Thangadurai: Harish-Chandra Research Institute, Department of Mathematics

Chapter Chapter 8 in Pillars of Transcendental Number Theory, 2020, pp 131-154 from Springer

Abstract: Abstract We begin with some basic tools necessary for the proof of Theorem 7.1.1 in Sect. 8.1. First, Theorem 7.1.1 is reduced to an equivalent statement; see Theorem 8.1.2. In Sect. 8.1.1, we derive a simple, but useful, non-trivial lower bound for a non-vanishing linear form in logarithms of algebraic numbers with bounded coefficients. Section 8.1.2 provides construction of an augmentative polynomial. In Sect. 8.1.3, we give the construction of the auxiliary polynomial $$\Phi (Z_0,\ldots ,Z_{n-1})$$ in several variables which generalises the function of a single complex variable employed by Gelfond. Basic estimates on $$\Phi $$ are shown in Sect. 8.1.4. The main difficulty is in the interpolation techniques. Usually the order of the derivatives is increased while leaving the points of interpolation fixed. Baker used a special extrapolation procedure in which the range of interpolation points is extended while the order of the derivatives is reduced, and the absolute values of these derivatives are shown to be very small. See Sects. 8.1.5 and 8.1.6.

Date: 2020
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DOI: 10.1007/978-981-15-4155-1_8

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