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

years.

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.