Let $X$ be the set of functions $e^{p(x)}$ of the real vector $x$, where $p$ is a multivariate polynomial with $p(0)=0$.
Is any finite subset of $X$ linearly independent? If yes, why? If no, is the answer true for other, restricted choices of $p$? (The answer is yes when the polynomials are restricted to have degree 1.)
Yes. If you require that no difference $p_j-p_k$ is constant (which follows from your assumption $p(0)=0$), then $c_1e^{p_1}+\ldots+c_me^{p_m}=0$ implies that all $c_j=0$.
In fact a more general result is true: instead of polynomials one can take any entire functions. This is called Borel's theorem. (See, for example, S. Lang, Introduction to complex hyperbplic spaces, Springer 1987, Theorem 1 in VII, sect 1, p. 186.)
For polynomials in one variable, this is easy: look at the asymptotics, for example when $z$ tends to infinity on an appropriate ray, and use induction in degree of the polynomials. To obtain the same for several variables, restrict your identity onto lines in $C^n$.
