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
of the
-th Hermite polynomial
, which is defined for integers
by

This is a classical orthogonal polynomial (which implies in particular that all zeros of
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}) x_n=\sqrt{2n}-\frac{i_1}{\sqrt[3]{6}}\frac{1}{(2n)^{1/6}}+o(n^{-1/6})](https://s0.wp.com/latex.php?latex=x_n%3D%5Csqrt%7B2n%7D-%5Cfrac%7Bi_1%7D%7B%5Csqrt%5B3%5D%7B6%7D%7D%5Cfrac%7B1%7D%7B%282n%29%5E%7B1%2F6%7D%7D%2Bo%28n%5E%7B-1%2F6%7D%29&bg=ffffff&fg=000000&s=0)
where
is the first (real) zero of the function

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
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

which is only asymptotically weaker by a factor
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
are real and simple, we will use the easy facts that
, and that
is an even function for
even, and an odd function for
odd, and most importantly (since all other properties are rather generic!) that they satisfy the differential equation

The crucial lemma is the following result of Laguerre:
Let
be a polynomial of degree
. Let
be a simple zero of
, and let

Then if
is any line or circle passing through
and
, either all zeros of
are in
, or both components of
contain at least one zero of
.
Before explaining the proof of this, let’s see how it gives the desired lower bound on the largest zero
of
. We apply Laguerre’s result with
and
. Using the differential equation, we obtain

Now consider the circle
such that the segment
is a diameter of
.

Now note that
is the smallest zero of
(as we observed above,
is either odd or even). We can not have
: if that were the case, the unbounded component of the complement of the circle
would not contain any zero, and neither would
contain all zeros (since
), contradicting the conclusion of Laguerre's Lemma. Hence we get 
and this implies

as claimed. (Note that if
, one deduces easily that the inequality is strict, but there is equality for
.)
Now for the proof of the Lemma. One defines a polynomial
by

so that
has degree
and has zero set
formed of the zeros of
different from
(since the latter is assumed to be simple). Using the definition, we have

We now compute the value at
of the logarithmic derivative of
, which is well-defined: we have

hence

which becomes, by the above formulas and the definition of
, the identity

or equivalently

where
is a Möbius transformation.
Recalling that
, this means that
is the average of the
. It is then elementary that for line
, either
is contained in
, or
intersects both components of the complement of
. Now apply
to this assertion: one gets that either
is contained in
, or
intersects both components of the complement of
. We are now done, after observing that the lines passing through
are precisely the images under
of the circles and lines passing through
and through
(because
, and each line passes through
in the projective line.)