  We compute the minimal and the maximal bound on the number of generators of a
minimal smooth monomial Togliatti system of forms of degree $d$ in $n+1$
variables, for any $d\ge 2$ and $n\geq 2$. We classify the Togliatti systems
with number of generators reaching the lower bound or close to the lower bound.
We then prove that if $n=2$ (resp $n=2,3$) all range between the lower and
upper bound is covered, while if $n\geq 3$ (resp. $n\ge 4$) there are gaps if
we only consider smooth minimal Togliatti systems (resp. if we avoid the
smoothness hypothesis). We finally analyze for $n=2$ the Mumford-Takemoto
stability of the syzygy bundle associated to smooth monomial Togliatti systems.
