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This article investigates when algebraic functions yield U_m-values for Liouville numbers, suggesting broader implications.
In 1953 LeVeque proved the existence of Uₘ U m -numbers by showing that for some specially defined Liouville number λ λ , the m th root λ 1/m λ 1 / m is in Uₘ U m . In this article we study the following question: let u be an algebraic function of degree m and λ λ a Liouville number; under which conditions is u(λ ) u ( λ ) a Uₘ U m -number? We consider a more refined notion of L L -numbers, and show that, under very general assumptions, an algebraic function of degree m takes Uₘ U m -values at all L L -numbers.
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Bilu et al. (2026) studied this question.
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