Minimal unitary representations from supersymmetry

We compute the supersymmetry constraints on the R-4 type corrections in maximal supergravity in dimension 8, 6, 4 and 3, and determine the tensorial differential equations satisfied by the function of the scalar fields multiplying the R-4 term in the corresponding invariants. The second-order derivative of this function restricted to the Joseph ideal vanishes in dimension lower than six. These results are extended to the del R-4(4) and the del R-6(4) corrections, based on the harmonic superspace construction of these invariants in the linearised approximation. We discuss the solutions of these differential equations and analysis the consequences on the non-perturbative type II low energy string theory effective action.