  A third-order three-dimensional symmetric traceless tensor, called the
\emph{octupolar} tensor, has been introduced to study tetrahedratic nematic
phases in liquid crystals. The octupolar \emph{potential}, a scalar-valued
function generated on the unit sphere by that tensor, should ideally have four
maxima capturing the most probable molecular orientations (on the vertices of a
tetrahedron), but it was recently found to possess an equally generic variant
with \emph{three} maxima instead of four. It was also shown that the
irreducible admissible region for the octupolar tensor in a three-dimensional
parameter space is bounded by a dome-shaped surface, beneath which is a
\emph{separatrix} surface connecting the two generic octupolar states. The
latter surface, which was obtained through numerical continuation, may be
physically interpreted as marking a possible \emph{intra-octupolar} transition.
In this paper, by using the resultant theory of algebraic geometry and the
E-characteristic polynomial of spectral theory of tensors, we give a
closed-form, algebraic expression for both the dome-shaped surface and the
separatrix surface. This turns the envisaged intra-octupolar transition into a
quantitative, possibly observable prediction. Some other properties of
octupolar tensors are also studied.
