If we have $n$ algebraic numbers $x_1,x_2,...,x_n$ $\in$ $\bar{\mathbb{Q}}^d$ which are linearly independent over $\mathbb{Q}$. How do we show that the $n$ functions
$f_i(z_1,z_2,...,z_d)= e^{{x_i}.\bar{z}} $ , $1 \leq n$
are algebraically independent over $\mathbb{Q}$.
Note that here product ${x_i}.\bar{z}$ is sum of the $d$-tuple after taking product component wise.
I have done for $d=1$ using the fact that if $x_1,x_2,...,x_n$ are distinct complex numbers then the functions $e^{x_iz}$ , $1 \leq i \leq n$ are linearly independent over $\mathbb{Q}$.