AI 中文总结
本文改进了Bessel函数$J_0$在代数点处对数导数值的超越度量，将指数从$4d^2\delta$降至更小的显式量$\mu(d,\delta)$，方法基于优化Siegel-Shidlovskii方法中的行列式大小。
AI 中文摘要
设$P\in \mathbb Z[X]\setminus\{0\}$为次数$\delta\ge 1$、通常高度$H\ge 1$的多项式，并设$\alpha\in \overline{\mathbb Q}^*$为次数$d\ge 2$的代数数。作为Lang和Galochkin一般性结果的一个推论，我们有以下超越度量：对任意$\varepsilon>0$，存在$c>0$使得$\vert P(J_0'(\alpha)/J_0(\alpha))\vert>c/H^{4d^2\delta+\varepsilon}$，其中$J_0$是Bessel函数。在本文中，我们证明指数$4d^2\delta$可以被一个更小的（显式）量$\mu(d,\delta)\le 4d^2\delta-2d\delta-1$所替代。类似的改进更一般地适用于任何微分阶为2且$f$与$f'$是齐次代数独立的$E$-函数$f$的对数导数。我们的方法基于对经典Siegel-Shidlovskii方法中自然出现的行列式大小的优化，遵循我们先前对任意$\alpha\in \overline{\mathbb Q}^*$的$e^\alpha$值的超越度量改进的步骤。
英文摘要
Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $δ\ge 1$ and usual height $H\ge 1$, and let $α\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. As a consequence of general result due to Lang and Galochkin, we have the following transcendence measure: for any $\varepsilon>0$, there exists $c>0$ such that $\vert P(J_0'(α)/J_0(α))\vert>c/H^{4d^2δ+\varepsilon}$ where $J_0$ is the Bessel function. In this paper, we prove that the exponent $4d^2δ$ can be replaced by a smaller (explicit) quantity $μ(d,δ)\le 4d^2δ-2dδ-1$. A similar improvement holds more generally for the logarithmic derivative of any $E$-function $f$ of differential order 2 and such that $f$ and $f'$ are homogeneously algebraically independent. Our method rests upon the optimization of the size of a determinant that appears naturally in the classical Siegel-Shidlovskii method, following the steps of our previous improvement of the transcendence measure of the value $e^α$ for any $α\in \overline{\mathbb Q}^*$.
