Effective Constructions in Transcendental Number Theory

Abstract

Many results in transcendental number theory are proved using the following principle. Suppose that we want to prove that a number α is irrational or transcendental. To do this, we would like to find a way to construct a sequence of sufficiently good Diophantine approximations to α — perhaps approximations by rational numbers (see §1.4 below), or else a sequence of simultaneous approximations (see §2), or possibly a sequence of polynomials with integer coefficients that take small values at α (see §4.2). If α were rational or algebraic, there would be some polynomial P(x) ∈ ℤ[x], P ≢ 0, having α as a root. If our sequence of approximations is “dense” enough, then we can use it to eliminate α in the equation P(α) = 0, thereby obtaining a contradiction.KeywordsLinear FormRational NumberHypergeometric FunctionAlgebraic NumberContinue Fraction ExpansionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.