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.) 



linear independence of exponentials
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 $C^n$.


