  The conditions for the existence of the effective action in statistical field
theory, the Legendre transform of the cumulant generating function, in presence
of non-linear local constraints are discussed. This problem is of importance
for non-perturbative approaches, such as the functional renormalization group.
We show that the Legendre transform exists as long as the non-linear
constraints do not imply linear constraints on the microscopic fields. We
discuss how to handle the case of effectively linear constraints and we
naturally obtain that the second derivative of the effective action is the
Moore-Penrose pseudo-inverse of the correlation function. We illustrate our
discussion with toy-models, and show that the correct counting of degrees of
freedom in non-linearly constrained statistical field theories can be rather
counter-intuitive.
