<p>The present paper deals with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2024_613_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-expansion in algebraic function fields. We characterize the set of elements in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2024_613_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_q(x,\beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> having purely periodic <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2024_613_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-expansions when <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2024_613_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> is a Pisot or Salem unit base by occurring their conjugates in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2024_613_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_q(x,\beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Purely periodic beta-expansions with Pisot or Salem unit series in $\mathbb {F} _{q}((x^{-1}))$

期刊论文
发表日期：

2025年1月7日

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R. Ghorbel,
S. Zouari

摘要

The present paper deals with $\beta $ $β$ -expansion in algebraic function fields. We characterize the set of elements in $\mathbb {F}_q(x,\beta )$ $F_{q} (x, β)$ having purely periodic $\beta $ $β$ -expansions when $\beta $ $β$ is a Pisot or Salem unit base by occurring their conjugates in $\mathbb {F}_q(x,\beta )$ $F_{q} (x, β)$ .

Purely periodic beta-expansions with Pisot or Salem unit series in \(\mathbb {F} _{q}((x^{-1}))\)

摘要