  We investigate the n-variable real functions G that are solutions of the
Chisini functional equation F(x)=F(G(x),...,G(x)), where F is a given function
of n real variables. We provide necessary and sufficient conditions on F for
the existence and uniqueness of solutions. When F is nondecreasing in each
variable, we show in a constructive way that if a solution exists then a
nondecreasing and idempotent solution always exists. We also provide necessary
and sufficient conditions on F for the existence of continuous solutions and we
show how to construct such a solution. We finally discuss a few applications of
these results.
