Joint Mathematics Colloquium
|If F is a polynomial of degree d, the Waring
rank of F is the least number of terms in an expression for F as a
linear combination of dth powers. For example, F = xy can be written as
a linear combination of squares:
xy = 1/4*(x+y)^2 - 1/4*(x-y)^2. The problems of finding the rank of a given polynomial and studying rank in general have applications throughout statistics, engineering, and the sciences, such as in signal processing and computational complexity, and they have been central problems 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.
I will describe joint work with J.M. Landsberg which gives a new, elementary improvement to Sylvester's catalecticant lower bound for the rank of a polynomial, in terms of the geometry of the polynomial. This is the first general bound for Waring rank since Sylvester.
This elementary talk will be accessible to general audiences with some background in undergraduate-level linear algebra and partial derivatives (as in multivariable calculus).