  We study the quantum theory of an O(N) scalar field on de Sitter geometry at
leading order in a nonperturbative 1/N-expansion. This resums the infinite
series of so-called superdaisy loop diagrams. We obtain the de Sitter symmetric
solutions of the corresponding, properly renormalized, dynamical field
equations and compute the complete effective potential. Because of its
self-interactions, the field acquires a strictly positive square mass which
screens potential infrared divergences. Moreover, strongly enhanced
ultralong-wavelength fluctuations prevent the existence of a spontaneously
broken symmetry state in any dimension.
