By the fundamental theorem of calculus we know a polynomial of degree $n$ has as many roots in the complex plane. That doesn't mean have any more information of the roots than that, (e.g. is the Galois group $S_n$?). There's also the problem that $e^z$ has *no* roots in $\mathbb{C}$ and how to account for that.

Using Stirling formula, Laplace transform, Rouché s theorem and things like that, one can so the roots of such polynomials behave like:
$$ |z \, e^{1-z} | = 1 $$
I don't know where I first learned of these results maybe in American Mathematical Monthly:

However, I later found many other sources. These polynomials appear whenever you have a holomorphic, or entire function or a Laplace transform or otherwise non-zero function and you need to take some kind of approximation.

This result is still counterintuitive, since setting $z = ix$ and observing $e^{i \pi x} \neq 0$ suggests there should never be zeros.

Varga and Carpenter show:

$$ \theta = |\arg z| \geq \cos^{-1} \left( 1 - \frac{2}{n}\right) $$

this should be enough to rule out a zero being on the imaginary axis.

Using the Cauchy formula, it's possible to write these truncated exponential series as an integral:

$$ (ez)^{-n} p_{n-1}(nz) = \frac{1}{2\pi i} \int_{\{|s|=1\}} \frac{s^{-n} \, e^{n(s-1)}}{s-z} \, ds$$

The exponential polynomials are in fact moments of a distribution.