Transcendence measures and algebraic growth of entire functions (Q2466353)

The paper deals with a transcendence measure of an entire function \(f\), which can be defined by \[ E_n=E_n(f)=\sup\{\| P\| _{\Delta^2}: P\in\mathbb{C}[z,w], \text{deg} P \leq n,\| P(z,f(z))\| _{\Delta}\leq 1\}, \] where \(\Delta\) is the closed unit disk in \(\mathbb{C},\quad \Delta^2=\Delta\times\Delta,\) and the norms are uniform norms. Let \(P_f(z)=P(z,f(z)),\quad \Delta_r=\{z\in \mathbb{C}: \;\;| z| \leq r\}.\) Denote by \[ e_n=\log E_n , \quad m(r)=m(r,f)=\max\{\log^+| f(z)| : \;| z| =r \}, \] \[ m_n(r)=m_n(r,f)=\sup\{\log\| P_f\| _{\Delta_r}: \text{deg} P \leq n,\| P_f \| _{\Delta}\leq 1 \}, \] and by \(Z_n(r)=Z_n(r,f)\) the maximum number of zeros of the functions \(P_f\) in the disk \(\Delta_r\) when \(\text{deg} P \leq n.\) The authors prove the following theorem. Theorem. For any entire function \(f\) of finite order \(\rho > 0,\) there exist sequences of integers \(\{n_j\}\) and \(\varepsilon_j > 0, \varepsilon_j \to 0,\) such that \[ e_{n_j}\leq C_1 n_j^2 \log n_j, \quad m_{n_j}(r)\leq C_2 n_j^2 \log r, \quad 1\leq r \leq \frac{1}{2}n_j^{1/\rho - \varepsilon_j}. \] For every \(r\geq 1\) there exists an integer \(j_r\) such that if \(j\geq j_r\) then \[ Z_{n_j}(r) \leq C_3 n_j^2, \quad \frac{M(2r,P_f)}{M(r,P_f)}\leq 2^{an_j^2},\quad M(r,P'_f) \leq C_4 n_j^2 \frac{M(r,P_f)}{r}, \] where \(P\) is a polynomial of degree at most \(n_j\) and \(M(r,F)= \max\{| F(z)| :\quad | z| =r \}.\) The constants \(C_j,\) \( a\) are effectively computed and depend exponentially only on \(\rho.\) The authors show that for special classes of functions, including the Riemann \(\xi-\)function, these estimates hold for all degrees of polynomials and are asymptotically best possible. From this theory they derive lower estimates for a certain algebraic measure of a set of values \(f(E)\), in terms of the size of the set \(E.\)