  The Nagata Conjecture is one of the most intriguing open problems in the area
of curves in the plane. It is easily stated. Namely, it predicts that the
smallest degree d of a plane curve passing through r $\ge$ 10 general points in
the projective plane P 2 with multiplicities at least l at every point,
satisfies the inequality d > $\sqrt$ r $\times$ l. This conjecture has been
proven by M. Nagata in 1959, if r is a perfect square greater than 9. Up to
now, it remains open for every non-square r $\ge$ 10, after more than a half
century of attention by many researchers. In this paper, we formulate new
transcendental versions of this conjecture coming from pluripotential theory
and which are equivalent to a version in C n of the Nagata Conjecture.
