Determining Whether a Multivariate Hyperexponential Function is Algebraic

Abstract

Let $F={\cal C}(x_1,x_2, \cdots, x_\ell, x_{\ell+1}, \cdots, x_m)$, where $x_1,x_2, \cdots, x_\ell$ are differential variables, and $x_{\ell+1}, \cdots, x_m$ are shift variables. We show that a hyperexponential function, which is algebraic over $F$, is of form

$$g(x_1,x_2, \cdots, x_m) q(x_1,x_2, \cdots, x_\ell)^\frac{1}{t} \omega_{\ell+1}^{x_{\ell+1}} \cdots \omega_m^{x_m}, $$

where $g \in F, q \in {\cal C}(x_1,x_2, \cdots, x_\ell), t \in {\cal Z}^+$ and $\omega_{\ell+1}, \cdots, \omega_m$ are roots of unity. Furthermore, we present an algorithm for determining whether a hyperexponential function is algebraic over $F$.