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<p>A general and effective criterion for determining the algebraic independence of formal power series in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_845_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {F}_q((T^{-1}))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, is still an active area of research, particularly when considering more complex functions or series expansions. In this note, our aim is to prove the algebraic independence of some series in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_845_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {F}_q((T^{-1}))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the analogue of Bundschuh’s criteria of algebraic independance in the fields of Laurent series.</p>

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ALGEBRAIC INDEPENDENCE OF SOME SERIES AND CONTINUED FRACTIONS IN THE FIELDS OF FORMAL POWER SERIES

  • 期刊论文

  • 发表日期：

    2025年8月14日

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  • Chiheb Ben Bechir

摘要

A general and effective criterion for determining the algebraic independence of formal power series in \(\mathbb {F}_q((T^{-1}))\) F q ( ( T - 1 ) ) , is still an active area of research, particularly when considering more complex functions or series expansions. In this note, our aim is to prove the algebraic independence of some series in \(\mathbb {F}_q((T^{-1}))\) F q ( ( T - 1 ) ) and the analogue of Bundschuh’s criteria of algebraic independance in the fields of Laurent series.