Spring 2016, Thursdays, Blocker 506A, 4:00-4:50 PM

Date: April 14

"  Symmetric border rank of the determinant and Young flattenings "
  Cameron Farnsworth  


In the late 1700's Edward Waring posed a question regarding writing natural numbers as sums of ${\scriptstyle d}$-th powers of natural numbers. Since the formulation of this problem many variants have been devised. In particular, any homogeneous complex degree ${\scriptstyle d}$ polynomial may be written as a linear combination of ${\scriptstyle d}$-th powers of linear forms. The symmetric rank or Waring rank of a homogeneous degree ${\scriptstyle d}$ polynomial ${\scriptstyle P}$ is the minimum number of terms needed to write a polynomial this way. The polynomial Waring problem asks for an explicit polynomial ${\scriptstyle P}$, what is its symmetric rank?

In this talk, we define a geometric variant of symmetric rank called symmetric border rank that allows us to work with the tools of algebraic geometry. We use the method of Young flattenings defined by Landsberg and Ottaviani to prove new lower bounds for the symmetric border rank of the polynomial ${\scriptstyle \det_n}$ obtained from the determinant of an ${\scriptstyle n\times n}$ matrix of indeterminates.