  Multiple root estimation problems in statistical inference arise in many
contexts in the literature. In the context of maximum likelihood estimation,
the existence of multiple roots causes uncertainty in the computation of
maximum likelihood estimators using hill-climbing algorithms, and consequent
difficulties in the resulting statistical inference.
  In this paper, we study the multiple roots phenomenon in maximum likelihood
estimation for factor analysis. We prove that the corresponding likelihood
equations have uncountably many feasible solutions even in the simplest cases.
For the case in which the observed data are two-dimensional and the unobserved
factor scores are one-dimensional, we prove that the solutions to the
likelihood equations form a one-dimensional real curve.
