Linear independence of values of G-functions
Linear independence of values of G-functions
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G 函数值的线性独立性
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发表时间：
2017
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通讯作者：
T. Rivoal
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作者：
S. Fischler;T. Rivoal
给定任意非多项式 $G$-function $F(z)=\sum\_{k=0}^\infty A\_k z^k$ 收敛半径 $R$，我们认为 $G$-函数 $F\_n^{[s]}(z)=\sum\_{k=0}^\infty \frac{A\_k}{(k+n)^s}z^k$ 对于任意整数 $s\geq 0$ 和 $n\geq 1$． 对于任何固定的代数数 $\alpha$ 这样 $0 \textless{} \vert \alpha \vert \textless{} R$ 和任意数字域 $\mathbb{K}$ 包含 $\alpha$ 还有 $A\_k$我们定义 $\Phi\_{\alpha, S}$ 作为 $\mathbb{K}$-由值生成的向量空间 $F\_n^{[s]}(\alpha)$， $n\ge 1$ 和 $0\leq s\leq S$． 我们证明 $u\_{\mathbb{K},F}\log(S)\leq \dim\_{\mathbb{K}}(\Phi\_{\alpha, S})\leq v\_F S$ 对于任何 $S$，有效常数 $u\_{\mathbb{K},F}\textgreater{}0$ 和 $v\_F\textgreater{}0$这就是家庭 $(F\_n^{[s]}(\alpha))\_{1\le n \le v\_F, s \ge 0}$ 包含无穷多个无理数。这个定理特别适用于 $F$ 是一个具有有理参数的超几何级数或多个多对数，它包含了第二作者和Marcovecchio在多对数情况下的先前结果。该证明依赖于一个明确的pad<e:1>型近似的构造。它利用了安德烈、丘德诺夫斯基和卡茨的研究结果 $G$-算子，一种新的线性无关准则（la Nesterenko）在数域上，奇点分析和鞍点法。
Given any non-polynomial $G$-function $F(z)=\sum\_{k=0}^\infty A\_k z^k$ of radius of convergence $R$, we consider the $G$-functions $F\_n^{[s]}(z)=\sum\_{k=0}^\infty \frac{A\_k}{(k+n)^s}z^k$ for any integers $s\geq 0$ and $n\geq 1$. For any fixed algebraic number $\alpha$ such that $0 \textless{} \vert \alpha \vert \textless{} R$ and any number field $\mathbb{K}$ containing $\alpha$ and the $A\_k$'s, we define $\Phi\_{\alpha, S}$ as the $\mathbb{K}$-vector space generated by the values $F\_n^{[s]}(\alpha)$, $n\ge 1$ and $0\leq s\leq S$. We prove that $u\_{\mathbb{K},F}\log(S)\leq \dim\_{\mathbb{K}}(\Phi\_{\alpha, S})\leq v\_F S$ for any $S$, with effective constants $u\_{\mathbb{K},F}\textgreater{}0$ and $v\_F\textgreater{}0$, and that the family $(F\_n^{[s]}(\alpha))\_{1\le n \le v\_F, s \ge 0}$ contains infinitely many irrational numbers. This theorem applies in particular when $F$ is an hypergeometric series with rational parameters or a multiple polylogarithm, and it encompasses a previous result by the second author and Marcovecchio in the case of polylogarithms. The proof relies on an explicit construction of Pad\'e-type approximants. It makes use of results of Andr\'e, Chudnovsky and Katz on $G$-operators, of a new linear independence criterion \`a la Nesterenko over number fields, of singularity analysis as well as of the saddle point method.