Roth’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 6 in Pillars of Transcendental Number Theory, 2020, pp 87-105 from Springer

Abstract: Abstract Thue’s and Siegel’s improvements of Liouville’s theorem depend on the construction of an auxiliary polynomial in two variables possessing zeros to a high order. Any further progress seemed to require non-trivial extension of the arguments relating to polynomials in several variables especially the possible multiplicities of its zeros. This was discovered by Roth in 1955, when he proved that $$\kappa $$ in Theorem 5.2.3 can be taken as $$2+\epsilon , \epsilon >0.$$ To deal with the multiplicities of zeros of multi-variable polynomials, Roth introduced the notion of index of a polynomial; see Sect. 6.1. This notion was later used by Vojta in 1991 in his proof of Falting’s famous theorem about Mordell conjecture.

Date: 2020
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DOI: 10.1007/978-981-15-4155-1_6

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