$ T$ is an algebraic polynomial $ $ T=p_0(x) + p_1(x) \cdot y + p_2(x)\cdot y^2 + \cdots + p_n(x)\cdot y^n $ $ $ p_i(x)$ is polynomial with rational coefficients. When $ T=0$ , or $ $ p_0(x) + p_1(x) \cdot y + p_2(x)\cdot y^2 + \cdots + p_n(x)\cdot y^n =0$ $

$ $ y=\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \ 0 & 0 &\cdots & 0 \ \end{array} \right)(\Omega)^4\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \ 0 & 0 &\cdots & 0 \ \end{array} \right)(\Omega)^4+\theta\left( \begin{array}{cccccc} 0 & 0 & \cdots & 0 \ 0 & 0 &\cdots & 0 \ \end{array} \right)(\Omega)^4\left( \begin{array}{cccccc} 0 & \frac{1}{2} & \cdots & 0 \ 0 & 0 &\cdots & 0 \ \end{array} \right)(\Omega)^4-\frac{\theta\left( \begin{array}{cccccc} 0 & 0 & \cdots & 0 \ \frac{1}{2} & 0 &\cdots & 0 \ \end{array} \right)(\Omega)^4\left( \begin{array}{cccccc} 0 & \frac{1}{2} & \cdots & 0 \ \frac{1}{2} & 0 &\cdots & 0 \ \end{array} \right)(\Omega)^4}{2\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \ 0 & 0 &\cdots & 0 \ \end{array} \right)(\Omega)^4\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \ 0 & 0 &\cdots & 0 \ \end{array} \right)(\Omega)^4}$ $ $ \theta$ is the Theta function, please see David Mumford Tate lecture on Theta Two Jacobian page 266 for more detail.

Now, my question is when $ y$ is expanded as power series (Taylor expansion) in $ x$ ,as $ y=\sum_0^{\infty }a_i x^i$ , under what condition $ a_i\in \mathbb{N}\bigcup 0$ ?