Here a log-concave sequence $(a_0,a_1,a_2,\ldots)$ is a sequence of positive real numbers such that $a_i^2 \geq a_{i-1}a_{i+1}$ for each $i\geq 1$. These are pervasive within mathematics.
A polynomial $p(x) = \sum_{i=0}^d a_i x^d$ is said to be log-concave if the sequence $(a_0,\ldots,a_d)$ of its coefficients is log-concave. We say that $p(x)$ is unimodal if there exists some index $j$ such that $a_0\leq \cdots\leq a_{j-1}\leq a_j\geq a_{j+1}\geq \cdots \geq a_d$. The following chain of implications is well known: $$ p(x) \text{ has only negative real roots} \Longrightarrow p(x)\text{ is log-concave} \Longrightarrow p(x) \text{ is unimodal}.$$
It is known that the usual product of polynomials has the following behavior:
- $p(x)$ and $q(x)$ are negative real-rooted $\Longrightarrow$ $p(x)q(x)$ is negative real-rooted.
- $p(x)$ and $q(x)$ are log-cocave $\Longrightarrow$ $p(x)q(x)$ is log-concave.
- $p(x)$ and $q(x)$ are unimodal does not necessarily imply that $p(x)q(x)$ is unimodal.
(The first one is trivial, and for a proof of the second and an example for the third one can look at Stanley's survey on unimodality and log-concavity in combinatorics). Here I am interested in obtaining analogous statements for a slightly more complicated notion of ``product'' of polynomials.
We define the (non-linear) operator $\mathscr{W}:\mathbb{R}[x]\to \mathbb{R}[x]$, that maps $f(x)\mapsto (\mathscr{W}f)(x)$, where $\mathscr{W}f$ is the only polynomial of degree $\leq \deg f(x)$ satisfying: $$\sum_{m=0}^{\infty} f(m) x^m = \frac{(\mathscr{W}f)(x)}{(1-x)^{1+\deg f(x)}}.$$ (This operator appears quite frequently in Ehrhart theory, combinatorics, commutative algebra, etc.)
In his PhD Thesis, David Wagner proved the following result (it corresponds to Theorem 0.3 in his paper Total positivity of Hadamard products)
Theorem: If $f$ and $g$ are polynomials such that $\mathscr{W}f$ and $\mathscr{W}g$ are negative real-rooted, then $\mathscr{W}(fg)$ is negative real-rooted too.
This is a quite important result in the theory of Pólya frequency sequences. It has many consequences in combinatorics, such as the validity of the Neggers-Stanley conjecture for certain classes of posets (though it fails for posets in general). My questions are the following:
Question 1: If $f$ and $g$ are polynomials such that $\mathscr{W}f$ and $\mathscr{W}g$ have positive coefficients and are log-concave, is it true that $\mathscr{W}(fg)$ is log-concave too?
Question 2: If $f$ and $g$ are polynomials such that $\mathscr{W}f$ and $\mathscr{W}g$ have positive coefficients and are unimodal, is it true that $\mathscr{W}(fg)$ is unimodal too?
My intuition is that the answer to the first question should be affirmative, and the answer to the second should be negative (of course, I haven't been able to produce any counterexamples). Unfortunately, the ideas in Wagner's proof for real-rootedness do not carry straightforwardly to the case of log-concavity.
