In 1953 LeVeque proved the existence of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$U_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation>-numbers by showing that for some specially defined Liouville number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>, the mth root <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda ^{1/m}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>λ</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>m</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> is in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$U_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation>. In this article we study the following question: let u be an algebraic function of degree m and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> a Liouville number; under which conditions is <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u(\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> a <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$U_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation>-number? We consider a more refined notion of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-numbers, and show that, under very general assumptions, an algebraic function of degree m takes <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$U_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation>-values at all <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-numbers.

Values of Algebraic Functions at Liouville Numbers

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2026年6月24日

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Yuri Bilu,
Diego Marques

摘要

In 1953 LeVeque proved the existence of $U_m$ $U_{m}$ -numbers by showing that for some specially defined Liouville number $\lambda $ $λ$ , the mth root $\lambda ^{1/m}$ $λ^{1 / m}$ is in $U_m$ $U_{m}$ . In this article we study the following question: let u be an algebraic function of degree m and $\lambda $ $λ$ a Liouville number; under which conditions is $u(\lambda )$ $u (λ)$ a $U_m$ $U_{m}$ -number? We consider a more refined notion of $\mathcal {L}$ $L$ -numbers, and show that, under very general assumptions, an algebraic function of degree m takes $U_m$ $U_{m}$ -values at all $\mathcal {L}$ $L$ -numbers.