  One question that we investigate in this paper is, how can we build
log-concave polynomials using sparse polynomials as building blocks? More
precisely, let $f = \sum\_{i = 0}^d a\_i X^i \in \mathbb{R}^+[X]$ be a
polynomial satisfying the log-concavity condition $a\_i^2 \textgreater{} \tau
a\_{i-1}a\_{i+1}$ for every $i \in \{1,\ldots,d-1\},$ where $\tau
\textgreater{} 0$. Whenever $f$ can be written under the form $f = \sum\_{i =
1}^k \prod\_{j = 1}^m f\_{i,j}$ where the polynomials $f\_{i,j}$ have at most
$t$ monomials, it is clear that $d \leq k t^m$. Assuming that the $f\_{i,j}$
have only non-negative coefficients, we improve this degree bound to $d =
\mathcal O(k m^{2/3} t^{2m/3} {\rm log^{2/3}}(kt))$ if $\tau \textgreater{} 1$,
and to $d \leq kmt$ if $\tau = d^{2d}$.
  This investigation has a complexity-theoretic motivation: we show that a
suitable strengthening of the above results would imply a separation of the
algebraic complexity classes VP and VNP. As they currently stand, these results
are strong enough to provide a new example of a family of polynomials in VNP
which cannot be computed by monotone arithmetic circuits of polynomial size.
