Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {F}_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> be a finite field of characteristic p and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\xi \in \mathbb {F}_{q}((T^{-1}))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>∈</mo> <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>. The approximation constant <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda _{n}(\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is defined as the supremum of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda \in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> such that the estimate <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\max _{0\le j\le k} \Vert Q\xi ^{k}\Vert \le |Q|^{-\lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo movablelimits="true">max</mo> <mrow> <mn>0</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">‖</mo> <mi>Q</mi> </mrow> <msup> <mi>ξ</mi> <mi>k</mi> </msup> <msup> <mrow> <mo stretchy="false">‖</mo> <mo>≤</mo> <mo stretchy="false">|</mo> <mi>Q</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>λ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> has infinitely many polynomial solutions Q. Here <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Vert .\Vert $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">‖</mo> <mo>.</mo> <mo stretchy="false">‖</mo> </mrow> </math></EquationSource> </InlineEquation> denotes the distance to the closest polynomial. In this paper, we determine the value of the Diophantine approximation exponent <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda _{n}(\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for hyperquadratic and non-hyperquadratic power series <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathbb {F}_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>.

Simultaneous diophantine approximation exponents of some algebraic power series in positive characteristic

期刊论文
发表日期：

2025年11月6日

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Khalil Ayadi,
Samir Elkadri

摘要

Let $\mathbb {F}_{q}$ $F_{q}$ be a finite field of characteristic p and $\xi \in \mathbb {F}_{q}((T^{-1}))$ $ξ \in F_{q} ((T^{- 1}))$ . The approximation constant $\lambda _{n}(\xi )$ $λ_{n} (ξ)$ is defined as the supremum of $\lambda \in \mathbb {R}$ $λ \in R$ such that the estimate $\max _{0\le j\le k} \Vert Q\xi ^{k}\Vert \le |Q|^{-\lambda }$ ${max}_{0 \leq j \leq k} ‖ Q ξ^{k} {‖ \leq | Q |}^{- λ}$ has infinitely many polynomial solutions Q. Here $\Vert .\Vert $ $‖ . ‖$ denotes the distance to the closest polynomial. In this paper, we determine the value of the Diophantine approximation exponent $\lambda _{n}(\xi )$ $λ_{n} (ξ)$ for hyperquadratic and non-hyperquadratic power series $\xi $ $ξ$ over $\mathbb {F}_{q}$ $F_{q}$ .