Understanding Umemura's Theorem for roots of algebraic equations

Question

I am trying to understand Umemura's Theorem for expressing the roots of any algebraic equation by higher genus theta functions. The original paper can be found here: Umemura, H.: Resolution of algebraic equations by theta constants. In: Mumford, D. (Ed.): Tata Lectures on Theta II, Birkhäuser, Boston/Basel/Stuttgart 1984. I apologize if the answers are quite straightforward for those familiar with the theorem.

R. Bruce King states the theorem as follows in his book "Beyond the Quartic Equation":

Let $f(x) = a_{0}x^{n} + a_{1}x^{n-1} + ... + a_{n} = 0$ with $a_{0} \neq 0$ be an algebraic equation irreducible over a certain subfield of the complex numbers. Then a root of this equation, $x_{k}$ can be expressed by the following equation involving theta

functions of zero argument:

$x_{k} = \left( \theta \left( \begin{array}{cccc} 1 & 0 & ... & 0 \\ 0 & ... & 0 & 0 \end{array} \right)(\Omega) \right)^{4} \left( \theta \left( \begin{array}{ccccc} 1 & 1 & 0 & ... & 0 \\ 0 & ... & 0 & 0 & 0 \end{array} \right)(\Omega) \right)^{4} + \left( \theta \left( \begin{array}{ccc} 0 & ... & 0 \\ 0 & ... & 0 \end{array} \right)(\Omega) \right)^{4} \left( \theta \left( \begin{array}{ccccc} 0 & 1 & 0 & ... & 0 \\ 0 & 0 & 0 & ... & 0\end{array} \right)(\Omega) \right)^{4} - \frac{\left( \theta \left( \begin{array}{cccc} 0 & 0 & ... & 0 \\ 1 & 0 & ... & 0 \end{array} \right)(\Omega) \right)^{4} \left( \theta \left( \begin{array}{ccccc} 0 & 1 & 0 & ... & 0 \\ 1 & 0 & ... & 0 & 0 \end{array} \right)(\Omega) \right)^{4} }{2 \left( \theta \left( \begin{array}{cccc} 1 & 0 & ... & 0 \\ 0 & ... & 0 & 0 \end{array} \right)(\Omega) \right)^{4} \left( \theta \left( \begin{array}{ccccc} 1 & 1 & 0 & ... & 0 \\ 0 & 0 & 0 & ... & 0 \end{array} \right)(\Omega) \right)^{4}}$\

where $\Omega$ is the period matrix derived from one of the following hyperelliptic integragls:

$u(a) = \int_1^a \frac{dx}{\sqrt{x(x-1)f(x)}}$ for odd degree of $f(x)$ and

$u(a) = \int_1^a \frac{dx}{\sqrt{x(x-1)(x-2)f(x)}}$ for even degree of $f(x)$.

The theta function is defined as follows:

For row vectors $m_{1}, m_{2} \in \mathbb{R}^g, z \in \mathbb{C}^g$ and a symmetric $g \times g$ matrix $\tau$ with positive definite imaginary part, we define the theta function $\theta \left( \begin{array}{c} m_{1} \\ m_{2} \end{array} \right)(z, \tau) = \Sigma_{\xi \in \mathbb{Z}^g} e(\frac{1}{2}(\xi + m_{1})\tau^{T}(\xi + m_{1}) + (\xi + m_{1})^{T}(z + m_{2}))$ where $e(x) = e^{2 \pi i x}$. We write $\theta \left( \begin{array}{c} m_{1} \\ m_{2} \end{array} \right)(\tau)$ for $\theta \left( \begin{array}{c} m_{1} \\ m_{2} \end{array} \right)(0, \tau)$.

My question about the definition are as follows: the equation gives a matrix, how can this be equal to a scalar root of an algebraic equation? The answer appears to be from Theorem 8.1 in the paper, but I can't understand it and would be grateful if someone could.

Finally, my question that may or may not be of mathematical interest is: can every root of an algebraic equation be found in this manner and can we then conclude that a number algebraic iff it satisfies Umemura's equation?

Answer 1

There is an obvious typo in the exponent: you need to transpose the first $(\xi + m_1)$. Then the exponent is a scalar, so theta function evaluates to a scalar as well.

I think $x_k$ is also a typo too. In my copy of Umemura's paper, the theorem is stated as follows.

Theorem 2. Let $$a_0X^n + a_1 X^{n-1} + \cdots + a_n = 0, a_0 \neq 0, a_i \in \mathbb{C}, (0 \leq i \leq n)$$ be an algebraic equation irreducible over a certain subfield of $\mathbb{C}$, then a root of the algebraic equation (above) is given (*), where $\Omega$ is the period matrix of a hyperelliptic curve $C: Y^2 = F(X)$ with $F(X) = X(X-1)(a_0X^n + a_1 X^{n-1} + \cdots + a_n)$ for $n$ odd and $F(X) = X(X-1)(X-2)(a_0X^n + a_1 X^{n-1} + \cdots + a_n)$ for $n$ even.

Here, (*) is the expression on the righthand side you typed for $x_k$. Note that the period matrix $\Omega$ is well-specified after you order the roots. In that paper, a specific order is given for this.