Zeros of MacLaurin polynomials for the exponential function

Question

Asked but never answered at MSE.

Let $\exp_n(z)$ denote the nth degree Taylor polynomial of $e^z$ :

$\exp_n(z) = 1 + z + z^2/2! + ... + z^n/n! \;$ .

The zeros of $\exp_n(z)$ were studied by Szego in the 1920's and later by others. One of the consequences of Szego's results is that the roots (after division by n) can come arbitrarily close to the imaginary axis.

Question: Is it possible for exp$_n$(z) to have a root that lies precisely on the imaginary axis?

Answer 1

Indeed the polynomial $\exp_n(z)$ has no purely imaginary zeros. Write $$ \exp_n(ix) = C_n(x) + i S_n(x) $$ in the obvious notation, with $C_n$ for truncations of cosine, and $S_n$ for truncations of sine. If there is a zero $ix$ for $\exp_n(z)$, then this $x$ must be a common root of $C_n$ and $S_n$, which are both polynomials in ${\Bbb Q}[x]$. Therefore $C_n$ and $S_n$ must have some non-trivial gcd in ${\Bbb Q}[x]$. Now we use the beautiful fact (going back to Schur) that $C_n(x)$ is irreducible, which completes our proof. I learnt of this result of Schur from these lovely notes of Keith Conrad; simply apply Theorem 1 there, or see Corollary 1 which mentions this explicitly.

Answer 2

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:

• 2003 On the Zeros of Truncated Exponential Series Peter Walker
• 2006 Zeros of sections of exponential sums Pavel Bleher, Robert Mallison
• blog Research in Scientific Computing in Undergraduate Education

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.

• 2007 Locating the zeros of partial sums of $\exp(z)$ with Riemann-Hilbert methods T. Kriecherbauer, A.B.J. Kuijlaars, K.D.T-R McLaughlin, P.D. Miller