$t^n=a$, we get one solution to the equation: $$t=e^{\frac{1}{n}\int^a_1 \frac{1}{x}}$$ generalizing this result by replacing the exponential with an elliptic modular function and the integral with hyperelliptic integrals, we can get a solution to an algebraic equation $a_0+a_1x+a_2x^2+\cdots+a_nx^n=0$ with degree above 5 by formulation of modular function and hyperelliptic integral(both formulated by Siegel Theta functions): $$x=\frac{\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}{2\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}+\frac{\theta\left( \begin{array}{cccccc} 0 & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} 0 & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}{2\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}-\frac{\theta\left( \begin{array}{cccccc} 0 & 0 & \cdots & 0 \\ \frac{1}{2} & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} 0 & \frac{1}{2} & \cdots & 0 \\ \frac{1}{2} & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}{2\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}$$
or
$$x=\frac{1}{2}+\frac{\theta\left( \begin{array}{cccccc} 0 & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} 0 & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}{2\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}-\frac{\theta\left( \begin{array}{cccccc} 0 & 0 & \cdots & 0 \\ \frac{1}{2} & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} 0 & \frac{1}{2} & \cdots & 0 \\ \frac{1}{2} & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}{2\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}$$ where $\Omega$ is the period matrix of the hyperelliptic curve $\mathbb{C}$, please see David Mumford Tate lecture on Theta $\textrm{II}$ Jacobian page 266 for more detail.
Now let us extend this result to algebraic equation with coefficients of $p_i(x) \in \mathbb{Q}[x]$ ,in other word, $p_i(x)$ is polynomial with coefficients of rational numbers.
$$p_0(x) + p_1(x) \cdot y + p_2(x)\cdot y^2 + \cdots + p_n(x)\cdot y^n $$ is an algebraic polynomial where $p_i(x)$ are the polynomials with rational coefficients. When $$p_0(x) + p_1(x) \cdot y + p_2(x)\cdot y^2 + \cdots + p_n(x)\cdot y^n =0$$, we have solution to the equation in which $y$ is formulated by modular function and hyperelliptic integrals with $x$ as variable, like $y= \phi(x)$, in another word, $$p_0(x) + p_1(x) \cdot \phi(x) + p_2(x)\cdot \phi(x)^2 + \cdots + p_n(x)\cdot \phi(x)^n =0$$
My question is when $y$ is expanded as power series (Taylor expansion) in $x$,as $$y=\sum_0^{\infty }b_i x^i$$, or $$\phi(x) = \sum_0^{\infty }b_i x^i$$, under what condition ( formulated by modular function and hyperelliptic integrals ) can we have $b_i\in \mathbb{N}\bigcup 0$, or can we find the condition under which the Taylor expansion of the function is series with coefficients of natural number?
