The minimal number of generators of a Togliatti system

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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≥ 2 and n≥ 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≥ 3 (resp. n≥ 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.