整函数的高度刚性
Height Rigidity for Entire Functions
AI总结：
本研究证明了有界次数且多项式有界高度的代数值在超越整函数图像的有理平移上是稀疏的，进而得到整函数取有理或有界次数代数值的刚性结果，排除了与Mahler关于刘维尔数问题相关的多项式分母情形，证明结合了Pila的有界次数计数定理等方法。
AI中文摘要：
我们证明，有界次数且多项式有界高度的代数值在超越整函数图像的有理平移上是稀疏的。更准确地说，对于固定的θ∈\u00afℚ∩ℝ、D≥1、t>0以及任意ε>0，高度至多为Q的有理数r∈[0,1]中，仅存在O(Q^ε)个能同时满足[ℚ(f(θ+r)):ℚ]≤D和H(f(θ+r))≪H(r)^t。因此，若这些界对所有足够高次的有理数r都成立，则f∈\u00afℚ[z]且deg f≤t。作为应用，我们得到了取有理或有界次数代数值且算术高度受多项式控制的整函数的刚性结果，这尤其排除了与Mahler关于刘维尔数问题自然相关的多项式分母情形。证明结合了Pila的有界次数计数定理、标准高度估计以及对超越整函数图像的简单几何分析。
英文摘要：
We prove that algebraic values of bounded degree and polynomially bounded height are sparse on rational translates of the graph of a transcendental entire function. More precisely, for fixed $θ\in\overline{\mathbb{Q}}\cap\mathbb{R}$, $D\geq1$, $t>0$, and for every $\varepsilon>0$, only $O(Q^{\varepsilon})$ rationals $r\in[0,1]$ of height at most $Q$ can satisfy simultaneously $[\mathbb{Q}(f(θ+r)):\mathbb{Q}]\leq D$ and $H(f(θ+r))\ll H(r)^t$. Consequently, if these bounds hold for every rational $r$ of sufficiently large height, then $f\in\overline{\mathbb{Q}}[z]$ and $°f\leq t$. As applications, we obtain rigidity results for entire functions taking rational or bounded-degree algebraic values with polynomially controlled arithmetic height. In particular, this excludes the polynomial-denominator scenario that arises naturally in connection with Mahler's problem on Liouville numbers. The proof combines Pila's bounded-degree counting theorem with standard height estimates and a simple geometric analysis of transcendental entire graphs.
