Abstract: The Waring rank of a polynomial of degree d 
is the least number of terms in an expression for the
polynomial as a sum of dth powers. The problem of
finding the rank of a given polynomial and studying
rank in general is related to secant varieties, and
there are applications throughout engineering and the
sciences, such as in signal processing and computational
complexity; and of course, it has been a central
problem of classical algebraic geometry. For example,
J.J. Sylvester gave a lower bound for rank in terms
of catalecticant matrices in the mid-19th century.
While catalecticant matrices and varieties have become
objects of study in their own right, there has been
relatively little progress in the last 150 years on
the problem of bounding or determining the rank of
a given (not general) polynomial, until the last 5-10

I will describe joint work with J.M. Landsberg which
gives an elementary improvement to the catalecticant
lower bound for the rank of a polynomial, in terms of
the geometry of the polynomial, with especially nice
results for some examples including plane cubic curves.