Ziming Li;Dabin Zheng
Journal of Systems Science and Complexity. 2006, 19(3): 352-364.
Let $F=\bC(x_1,x_2, \cdots, x_\ell, x_{\ell+1}, \cdots,
x_m)$, 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
\[
g(x_1,x_2, \cdots, x_m) q(x_1,x_2, \cdots, x_\ell)^\frac{1}{t}
\om_{\ell+1}^{x_{\ell+1}} \cdots \om_m^{x_m},
\]
where $g \in F$, $q \in \bC(x_1,x_2, \cdots, x_\ell)$, $t \in
\bZ^+$ and $\om_{\ell+1}, \cdots, \om_m$ are roots of unity.
Furthermore, we present an algorithm for determining whether a
hyperexponential function is algebraic over $F$.
