Extensions Due to Ramachandra

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 4 in Pillars of Transcendental Number Theory, 2020, pp 45-60 from Springer

Abstract: Abstract In 1968, Ramachandra [1, 2] proved results relating to the set of complex numbers at which a given set of algebraically independent meromorphic functions assumes values in a fixed algebraic number field. These results proved to be significant in the case, to quote his own words “(overlooked by Gelfond) where the functions concerned do not satisfy algebraic differential equations of the first order with algebraic number coefficients.” His result, besides simplifying Schneider’s method, enables one to study the set of all complex numbers at which two algebraically independent meromorphic functions f(z) and g(z) take values which are algebraic numbers. In particular, he was able to obtain results when $$(f(z), g(z))\in \{(z,\wp (az)),(e^z,\wp (az)),(\wp _1(z),\wp _2(az))\}$$ where $$a\ne 0$$ is an arbitrary complex number and $$\wp ,\wp _1$$ and $$\wp _2$$ are Weierstrass elliptic functions. We refer to [2] for these results.

Date: 2020
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DOI: 10.1007/978-981-15-4155-1_4

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