Claremont Center for the Mathematical Sciences

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 d-th powers. The problem of finding the rank of a given polynomial and studying rank in general has been a central problem of classical algebraic geometry, related to secant varieties; in addition, there are applications to statistics, signal processing, and computational complexity. Other than a well-known lower bound for rank in terms of catalecticant matrices, there has been relatively little progress on the problem of determining or bounding rank for a given polynomial (although related questions have proved very fruitful). I will describe new upper and lower bounds. The improved lower bound is especially interesting, dealing with the geometry of catalecticants. The talk should be accessible to all. This is joint work with J.M. Landsberg.