p-adic versions of log concavity for graphs (or matroids) It was recently shown using techniques inspired by algebraic geometry (by Huh and Adiprasito-Huh-Katz) that the chromatic polynomial of a graph (or matroid) has coefficients that satisfy log-concavity. Since the coefficients are integers, we can also ask if they satisfy any p-adic identities or inequalities. Are there any conjectures or results in this direction?
 A: I am not aware of any conjectures or results in this direction,
and I am not so optimistic because it seems to be a hard question even for binomial coefficients.
So I think your suggestion of studying coefficients of polynomials of the form
$\prod_i (t - n_i)$ is good, but I don't know what to expect.
Some references on $p$-adic properties of binomial coefficients:

*

*Wikipedia, divisibility properties of binomial coefficients.


*Erdos, Graham, Ruzsa, Straus, On the prime factors of $\binom{2n}{n}$.


*Spiegelhofer and Wallner, Divisibility of binomial coefficients by powers of 2.

We get binomial coefficients from taking the chromatic polynomial
of a tree with $n$ edges, which is
$$
\chi_\text{tree}(t) = t(t-1)^n = \sum_{k=0}^n (-1)^k  \binom{n}{k} t^{n-k+1} .
$$
The binomial coefficients are unimodal in the archimedean norm, but are far from unimodal $p$-adically. In fact, if $n$ is even, then the $2$-valuation of the sequence $\binom{n}{0}, \binom{n}{1}, \binom{n}{2},\ldots$ will alternate between getting larger and smaller, so in a sense behaves completely "opposite" to being unimodal.
At another extreme,
if $n = 2^m - 1$, then the binomial coefficients $\binom{n}{k}$ are all odd, so have the same size $2$-adically.
The binomial coefficients have a nice archimedean limit as $n \to \infty$, up to an appropriate rescaling, but the $p$-adic limit seems more complicated.
For instance, in the paper above Spiegelhofer and Wallner show that the 2-valuations of the binomial coefficients do obey a close-to normal distribution, but in the sense where we ignore the order of the sequence $\binom{n}{0}, \binom{n}{1}, \binom{n}{2}, \ldots$ and only count how many times numbers of a given $p$-adic size appear.

Another reason it seems challenging to find a proper $p$-adic analogue:
Adiprasito-Huh-Katz showed that the log-concavity in the archimedean norm comes from
the Hodge-Riemann relation in degrees 0 and 1 (applied to the Chow ring of a graph / matroid),
which says a certain matrix with integer entries must have indefinite signature.
Namely, log concavity of $a_{i-1}, a_i, a_{i+1}$
is equivalent to
$$
 a_{i-1}a_{i+1} \leq a_i^2
\quad\Leftrightarrow\quad
\det \begin{pmatrix}
a_{i-1} & a_i \\
a_i & a_{i+1}
\end{pmatrix} \leq 0.
$$
So what's important is not exactly the valuation $x \mapsto |x|$ that measures the size of a number,
but the fact that the integers (and reals) are totally ordered so that it makes sense to distinguish positive and negative numbers.
So one approach may be to ask:
Is there any reason to expect a $p$-adic version of the Hodge-Riemann relation? (I have no idea..)
Generally, is there a $p$-adic analogue of the signature of a bilinear form?
