  Koiran showed that if a $n$-variate polynomial of degree $d$ (with
$d=n^{O(1)}$) is computed by a circuit of size $s$, then it is also computed by
a homogeneous circuit of depth four and of size
$2^{O(\sqrt{d}\log(d)\log(s))}$. Using this result, Gupta, Kamath, Kayal and
Saptharishi gave a $\exp(O(\sqrt{d\log(d)\log(n)\log(s)}))$ upper bound for the
size of the smallest depth three circuit computing a $n$-variate polynomial of
degree $d=n^{O(1)}$ given by a circuit of size $s$.
  We improve here Koiran's bound. Indeed, we show that if we reduce an
arithmetic circuit to depth four, then the size becomes
$\exp(O(\sqrt{d\log(ds)\log(n)}))$. Mimicking Gupta, Kamath, Kayal and
Saptharishi's proof, it also implies the same upper bound for depth three
circuits.
  This new bound is not far from optimal in the sense that Gupta, Kamath, Kayal
and Saptharishi also showed a $2^{\Omega(\sqrt{d})}$ lower bound for the size
of homogeneous depth four circuits such that gates at the bottom have fan-in at
most $\sqrt{d}$. Finally, we show that this last lower bound also holds if the
fan-in is at least $\sqrt{d}$.
