# E. Kowalski’s blog

## Zeros of Hermite polynomials

In my paper with É. Fouvry and Ph. Michel where we find upper bounds for the number of certain sheaves on the affine line over a finite field with bounded ramification, the combinatorial part of the argument involves spherical codes and the method of Kabatjanski and Levenshtein, and turns out to depend on the rather recondite question of knowing a lower bound on the size of the largest zero $x_n$ of the $n$-th Hermite polynomial $H_n$, which is defined for integers $n\geq 1$ by
$H_n(x)=(-1)^n e^{x^2} \frac{d^n}{dx^n}e^{x^2}.$

This is a classical orthogonal polynomial (which implies in particular that all zeros of $H_n$ are real and simple). The standard reference for such questions seems to still be Szegö’s book, in which one can read the following rather remarkable asymptotic formula:
$x_n=\sqrt{2n}-\frac{i_1}{\sqrt[3]{6}}\frac{1}{(2n)^{1/6}}+o(n^{-1/6})$
where $i_1=3.3721\ldots>0$ is the first (real) zero of the function
$\mathrm{A}(x)=\frac{\pi}{3}\sqrt{\frac{x}{3}}\Bigl\{J_{1/3}\Bigl(2\Bigl(\frac{x}{3}\Bigr)^{3/2}\Bigr)+J_{-1/3}\Bigl(2\Bigl(\frac{x}{3}\Bigr)^{3/2}\Bigr)\Bigr\}$
which is a close cousin of the Airy function (see formula (6.32.8) in Szegö’s book, noting that he observes the Peano paragraphing rule, according to which section 6.32 comes before 6.4).

(Incidentally, if — like me — you tend to trust any random PDF you download to check a formula like that, you might end up with a version containing a typo: the cube root of $6$ is, in some printings, replaced by a square root…)

Szegö references work of a number of people (Zernike, Hahn. Korous, Bottema, Van Veen and Spencer), and sketches a proof based on ideas of Sturm on comparison of solutions of two differential equations.

As it happens, it is better for our purposes to have explicit inequalities, and there is an elementary proof of the estimate
$x_n\geq\sqrt{\frac{n-1}{2}},$
which is only asymptotically weaker by a factor $2$ from the previous formula. This is also explained by Szegö, and since the argument is rather cute and short, I will give a sketch of it.

Besides the fact that the zeros of $H_n$ are real and simple, we will use the easy facts that $\deg(H_n)=n$, and that $H_n$ is an even function for $n$ even, and an odd function for $n$ odd, and most importantly (since all other properties are rather generic!) that they satisfy the differential equation
$y''-2xy'+2ny=0.$

The crucial lemma is the following result of Laguerre:

Let $P\in \mathbf{C}[X]$ be a polynomial of degree $n\geq 1$. Let $z_0$ be a simple zero of $P$, and let
$w_0=z_0-2(n-1)\frac{P'(z_0)}{P''(z_0)}.$
Then if $T\subset \mathbf{C}$ is any line or circle passing through $z_0$ and $w_0$, either all zeros of $P$ are in $T$, or both components of $\mathbf{C}-T$ contain at least one zero of $P$.

Before explaining the proof of this, let’s see how it gives the desired lower bound on the largest zero $x_n$ of $H_n$. We apply Laguerre’s result with $P=H_n$ and $z_0=x_n$. Using the differential equation, we obtain
$w_0=x_n-\frac{n-1}{x_n}.$
Now consider the circle $T$ such that the segment $[w_0,z_0]$ is a diameter of $T$.

Now note that $-x_n$ is the smallest zero of $H_n$ (as we observed above, $H_n$ is either odd or even). We can not have $w_0<-x_n$: if that were the case, the unbounded component of the complement of the circle $T$ would not contain any zero, and neither would $T$ contain all zeros (since $-x_n\notin T$), contradicting the conclusion of Laguerre's Lemma. Hence we get $-x_n\leq w_0=x_n-\frac{n-1}{x_n},$
and this implies
$x_n\geq \sqrt{\frac{n-1}{2}},$
as claimed. (Note that if $n\geq 3$, one deduces easily that the inequality is strict, but there is equality for $n=2$.)

Now for the proof of the Lemma. One defines a polynomial $Q$ by
$P=(X-z_0)Q,$
so that $Q$ has degree $n-1$ and has zero set $Z$ formed of the zeros of $P$ different from $z_0$ (since the latter is assumed to be simple). Using the definition, we have
$Q'(z_0)=P'(z_0),\quad\quad Q''(z_0)=\frac{1}{2}P''(z_0).$
We now compute the value at $z_0$ of the logarithmic derivative of $Q$, which is well-defined: we have
$\frac{Q'}{Q}=\sum_{\alpha\in Z}\frac{1}{X-\alpha},$
hence
$\frac{Q'}{Q}(z_0)=\sum_{\alpha\in Z}\frac{1}{z_0-\alpha},$
which becomes, by the above formulas and the definition of $w_0$, the identity
$\frac{1}{z_0-w_0}=\frac{1}{n-1}\sum_{\alpha\in Z}\frac{1}{z_0-\alpha},$
or equivalently
$\gamma(w_0)=\frac{1}{n-1}\sum_{\alpha\in Z}{\gamma(\alpha)},$
where $\gamma(z)=1/(z_0-z)$ is a Möbius transformation.

Recalling that $|Z|=n-1$, this means that $\gamma(w_0)$ is the average of the $\gamma(\alpha)$. It is then elementary that for line $L$, either $\gamma(Z)$ is contained in $L$, or $\gamma(Z)$ intersects both components of the complement of $L$. Now apply $\gamma^{-1}$ to this assertion: one gets that either $Z$ is contained in $\gamma^{-1}(L)$, or $Z$ intersects both components of the complement of $\gamma^{-1}(L)$. We are now done, after observing that the lines passing through $\gamma(w_0)$ are precisely the images under $\gamma$ of the circles and lines passing through $w_0$ and through $z_0$ (because $\gamma(z_0)=\infty$, and each line passes through $\infty$ in the projective line.)

Written by Kowalski

February 20th, 2013 at 10:43 pm

Posted in Exercise,Mathematics

## Am I a topologist?

Topological thinking is rooted in local issues — the soil, the plants, the weather, and the local customs. Peoples’ well-being is its objective.

Written by Kowalski

February 14th, 2013 at 8:56 pm

“Orismology”, from the Oxford English Dictionary Word Of The Day, is the right term for the discussion of technical terminology (the théorie des termes de métier, as we say in French).

Written by Kowalski

February 7th, 2013 at 1:55 pm

Posted in Language,Mathematics

## Stickelberger’s copy of Jacobi’s “Canon arithmeticus”

I am currently the head of the Mathematics Library at ETH (which is separate from the main library). A few days ago, I surveyed some of the (relatively) old books in our collection with one of the librarians, just to see if some of these should be handled in a special way. We didn’t find anything really out of the ordinary (no copy of Poincaré’s works heavily annotated by H. Weyl, I’m afraid), but one book has some historical interest: it is (or seems to be) Stickelberger’s copy of Jacobi’s “Canon arithmeticus”

a table of primitive roots and discrete logarithms for primes up to 1000.

Stickelberger’s signature is found on one of the first pages

The table itself, as it took me a few minutes to understand (my Latin being non-existent), lists for each prime $p\leq 1000$ the “Numeri” $1\leq N\leq p-1$ and the “Index” $1\leq I\leq p-1$, which are defined by the relation
$N=\rho^I,$
for some primitive root $\rho$ modulo $p$, which can be identified easily by looking for the number for which $I$ is equal to $1$:
$\rho=\rho^1.$

So above we see that Jacobi selected $2$ as primitive root modulo $5$ and $11$, and $3$ as primitive root modulo $7$, or $6$ as primitive root modulo $13$. Obligingly, he also indicates the factorization of $p-1$ (so that all primitive roots can be easily found by checking whether the corresponding index is coprime with $p-1$).

Like the copy which was digitized by Google, Stickelberger’s has a list of corrections at the end, and most (if not all: I didn’t check…) of these are incorporated in pencil in the main text, as here with $p=71$:

However, Stickelberger (if it was him) also had another list of corrections, written down on a separate loose sheet of paper inserted at the end of the book.

These corrections are reproduced from the paper On quasi-mersennian numbers by Lieutenant Colonel Allan Cunningham in Vol. 41 of the Messenger of Mathematics (a volume which seems famous in statistical circles because it contains, ten pages later, an important paper of R.A. Fisher on maximum likelihood…) But even Cunningham’s corrections contain a few mistakes, which Stickelberger reports (though with question marks):

Indeed, for $p=757$, the primitive root chosen by Jacobi is $\rho=2$ and we have
$2^{468}=565\bmod{757},$
instead of $568$ reported by Cunningham (and $168$ in the Canon).

As far as I could see during my quick inspection, there are no further annotations or comments by Stickelberger, nor any date indicating when he acquired this book. The publication date is 1839, and the only other indication is that the volume of the Messenger of Mathematics with Cunningham’s paper appeared in 1912. I also do not known when and how the book entered the collection of ETH.

Written by Kowalski

February 3rd, 2013 at 5:45 pm

## A cruel dilemma

with one comment

From a recent article in the New York Times:

“On the satellite channels, I watch ‘America’s Got Talent’ dubbed in Persian, while at the same time, our state television is showing an hourlong program on mathematics. Which one would you prefer?” asked ************, 30, an insurance salesman.

Written by Kowalski

January 22nd, 2013 at 11:05 am

Posted in Mathematics