On the distance
between zeros of Taylor polynomials of the exponential function

The exponential function of a complex variable $z$ can be written as the infinite series \[ e^z = \sum_{k=0}^{\infty} \frac{z^k}{k!}. \] We can then let $p_{n}(z)$ denote the n

We can plot the $n$ complex zeros of $p_n(z)$. Here's a plot for $n = 44$ with the unit circle $|z| = 1$ shown in blue for reference.

(The Mathematica code used to produce this image can be found at the bottom of the page.)

These polynomials (and their zeros) have been studied for quite some time. In 2005 Stephen M. Zemyan published a very nice paper titled "On the Zeroes of the Nth Partial Sum of the Exponential Series" in which he gave an overview of some of the main results in this area and proved several new ones. He also posed a number of problems, and we aim to address two of those problems here.

Zemyan's problems

Section 4 of Zemyan's paper explores the spacing of the zeros of $p_n(z)$.

Looking at the plot above, we can see that the zeros on the left are closer together than the zeros on the right. For each $n \geq 2$, Zemyan defined the number $D_n$ to be the smallest distance between the zeros of $p_n(z)$.

For example, the zeros of the polynomial $p_2(z) = 1 + z + z^2/2$ are $z = -1 \pm i$, so the distance between the zeros is $2$, and hence $D_2 = 2$.

Using numerical methods we can calculate $D_n$ for larger $n$, and Zemyan includes the first six in his paper: \[ D_2 = 2, \quad D_3 \approx 2.016408, \quad D_4 \approx 1.77948, \] \[ D_5 \approx 1.775241, \quad D_6 \approx 1.676700, \quad D_7 \approx 1.672533. \]

Zemyan notes that $D_n$ seems to decrease for $n \geq 3$, and in Theorem 4 of his paper gives an explicit lower bound:

He then poses two problems relating to the spacing of the zeros (Problem 3 and Problem 4 in the paper):

Problem 4 restates what we noticed earlier about the zeros, that the ones on the left are closer together than the ones on the right.

We don't aim to give completely rigorous solutions to these two problems. Instead we will explore them from an asymptotic perspective, tackling them with approximations rather than exact formulae. To do this we'll first need to piece together a few known facts about the polynomials $p_n(z)$.

The Szegő curve

As Zemyan notes in his paper, Szegő showed in 1924 that the zeros of $p_n(z)$ grow proportionally to $n$ as $n$ increases. Even more, Szegő showed that after scaling all the zeros by $1/n$, as $n \to \infty$ they converge to the curve defined by \[ \left| ze^{1-z} \right| = 1, \quad |z| \leq 1. \] This curve has come to be known as the Szegő curve.

Here's a plot of the zeros of $p_{44}(45z)$ with the Szegő curve shown in red and the unit circle shown in blue. Compare this with the plot of the zeros of $p_{44}(z)$ at the top of the page.

Let's call the Szegő curve $S$ for short.

There are two points of $S$ which are of particular interest: the corner on the right, and the base of the bulb all the way on the left. The corner is located at $z=1$, but it's a little trickier to find where the base of the bulb is located.

The base lies on the real axis, so we'll write $z=x$ to indicate that it's a real variable. We also note that $x$ is negative. By using these facts in the definition of $S$ we get that \[ 1 = \left| ze^{1-z} \right| = -xe^{1-x}, \] and so \[ -xe^{-x} = 1/e. \] Calling on our friend the Lambert W function, we can solve this equation to get $-x = W(1/e)$ and hence $x = -W(1/e)$. We have proved the following Lemma.

We will also need some way to approximate the zeros. The following lemma is a bit technical but very cool. The limit in it was essentially first obtained by Edrei, Saff, and Varga in their monograph Zeros of Sections of Power Series. A proof of a more general version of it can be found in chapter 4 of my PhD thesis. I may eventually write another note here giving a direct proof of this specific case, but for now let's just accept the result and see where it gets us.

How can we interpret this? The main idea is that the argument $z_n(w)$ which gets plugged into the polynomial $p_{n-1}(z)$ is designed so that it "tracks" with the zeros of $p_{n-1}(z)$ as $n$ grows. We can illustrate this with an animation.

In this animation we show the zeros of $p_{n-1}(z)$ as $n$ increases from $20$ to $100$. We also draw a rectangle indicating the range of $z_n(w)$ as $w$ ranges over the square with corners at $\pm 15 \pm 15i$.

We can see that the square follows along with the zeros, staying at roughly the same place on the arc. The number of zeros in the square is eventually constant, and the zeros in the square gradually flatten out into a straight line. What Lemma 2 is saying is that, inside any moving square like this one, the partial sums $p_{n-1}(z)$ look a lot like the function $1-Ce^{-w}$ for some constant $C$, so we can approximate the zeros of the former with the zeros of the latter. We just need to plug the zeros of $1-Ce^{-w}$ back into the map $z_n(w)$ to get approximations for the zeros of $p_{n-1}(z)$.

Here's a plot where we've chosen an arbitrary point $\xi$ (shown in blue) on the Szegő curve $S$ and used Lemma 2 to obtain approximations for the zeros of $p_{44}(45z)$ near $\xi$. The approximations are shown as blue crosses. Near $\xi$ the approximations are very good.

We've gathered all the necessary tools so let's get back to Zemyan's problems.

Addressing the
problems

Let $\xi$ be any point of the Szegő curve $S$ except for its corner (so $\xi \neq 1$). Using Lemma 2, if $w_1$ and $w_2$ are solutions to the equation \[ 1 - \frac{e^{-w}}{(1-\xi)\sqrt{2\pi}} = 0, \tag{$*$} \] then $p_{n-1}(z)$ has zeros of the form \[ z = z_n(w_1) + \epsilon_1(n) \qquad \text{and} \qquad z = z_n(w_2) + \epsilon_2(n), \] where $\epsilon_1(n) \to 0$ and $\epsilon_2(n) \to 0$ as $n \to \infty$.

The distance between adjacent roots of equation $(*)$ is $2\pi$, so the limiting distance between the adjacent zeros of $p_{n-1}(z)$ which are closest to $n\xi$ is \[ \begin{align} |z_n(w_1) - z_n(w_2)| &= \left| n\xi \left(1 + \frac{\log n}{2(1-\xi)n} - \frac{w_1-i\tau_n}{(1-\xi)n}\right) - n\xi \left(1 + \frac{\log n}{2(1-\xi)n} - \frac{w_2-i\tau_n}{(1-\xi)n}\right) \right| \\ &= |w_1 - w_2| \left|\frac{\xi}{1-\xi} \right| \\ &= 2\pi \left|\frac{\xi}{1-\xi} \right|. \tag{$**$} \end{align} \] If $\xi \in S$ then $|\xi| = e^{\operatorname{Re} \xi - 1}$, so $|\xi|$ increases as $\xi$ moves from the leftmost point of $S$ at $\xi = -W(1/e)$ toward the corner at $\xi = 1$, while $|1 - \xi|$ decreases. Thus the limiting distance between adjacent zeros, $2\pi|\xi/(1-\xi)|$, increases as the real part of $\xi$ increases and is minimized when $\xi = -W(1/e)$.

Zemyan's Problem 4 asks whether the distance between adjacent zeros increases as the real parts of the zeros increase, and we have shown that this is, indeed, asymptotically true.

Zemyan's Problem 3 asks about the limit of $D_n$, the minimum distance between zeros of $p_n(z)$. Equation $(**)$ and the above argument yield the following result.

As a sanity check, this value is consistent with the values of $D_2, \ldots, D_7$ and Zemyan's lower bound. Even further, numerical computation yields $D_{100} \approx 1.399862$ and $D_{200} \approx 1.385674$.

If anyone has made progress on any of Zemyan's other problems I would love to hear about it. It would be nice to maintain a small repository of information about them.

Update Feb 8, 2018: The value of $D_{100}$ at the end was computed incorrectly. This page has been updated with the correct value.

Update Mar 29, 2018: The value of $D_{200}$ was also computed incorrectly. I've recomputed $D_{100}$ and $D_{200}$ using a different method and I'm pretty sure they're right now.

The Mathematica notebook (.nb) which I used to create the images in this note can be downloaded here.

Antonio R. Vargas

February 7, 2018