  We consider the notions of (i) critical points, (ii) second-order points,
(iii) local minima, and (iv) strict local minima for multivariate polynomials.
For each type of point, and as a function of the degree of the polynomial, we
study the complexity of deciding (1) if a given point is of that type, and (2)
if a polynomial has a point of that type. Our results characterize the
complexity of these two questions for all degrees left open by prior
literature. Our main contributions reveal that many of these questions turn out
to be tractable for cubic polynomials. In particular, we present an
efficiently-checkable necessary and sufficient condition for local minimality
of a point for a cubic polynomial. We also show that a local minimum of a cubic
polynomial can be efficiently found by solving semidefinite programs of size
linear in the number of variables. By contrast, we show that it is strongly
NP-hard to decide if a cubic polynomial has a critical point. We also prove
that the set of second-order points of any cubic polynomial is a spectrahedron,
and conversely that any spectrahedron is the projection of the set of
second-order points of a cubic polynomial. In our final section, we briefly
present a potential application of finding local minima of cubic polynomials to
the design of a third-order Newton method.
