Straight line complexity of monomials Let $k$ be some field. As usual, for an $f\in k[x_{1},x_{2},\ldots,x_{n}]$
we define $L(f)$ to be the straight-line complexity of $f$ over
$k$. Let $F$ be the set of monomials of $f$, namely the monomials
which appear in $f$ with non-zero coefficient. 


  Is it true that $ \forall m\in F:L(m)\le L(f)$?


Even some weaker upper bound for $L(m)$ is known?
If $$f=(\Sigma_{i=1}^n x_i)^{2^n}$$
then it has ${2^n+n-1\choose n-1} \approx 2^{n^2}$ monomials and $L(f)=O(n)$.
By a counting argument, there are $2^{O(n\log n)}$ straight-line programs of length $O(n)$.
As $f$ has more monomials, for some we need a longer program.
In fact this argument gives a monomial $m$ for which $L(m)=\widetilde\Omega(L^2(f))$.