Determining Whether a Multivariate Hyperexponential Function is Algebraic

Let F = C(x1, . . . , x`, x`+1, . . . , xm), where x1, . . . , x` are continuous variables, and x`+1, . . . , xm are discrete variables. We show that a hyperexponential function, which is algebraic over F , is of form g(x1, . . . , xm)q(x1, . . . , x`) 1 t ω x`+1 j+1 · · ·ωxm m where g ∈ F , q ∈ C(x1, . . . , x`), t ∈ Z and ω`+1, . . . , ωm are roots of unity. Furthermore, we present an algorithm for determining whether a hyperexponential function is algebraic over F .