Farnsworth, Cameron Lee (2016-08). The Polynomial Waring Problem and the Determinant. Doctoral Dissertation.
The symmetric rank of a polynomial P is the minimum number of d-th powers of linear forms necessary to sum to P. Questions pertaining to the rank and decomposition of symmetric forms or polynomials are of classic interest. Work on this topic dates back to the mid 1800's to J. J. Sylvester. Many questions have been resolved since Sylvester's work, yet many more questions have arisen. In recent years, certain polynomials including detn, the determinant of an n x n matrix of indeterminates, have become central in the study of rank problems. Symmetric border rank of a polynomial P is the minimum r such that P is in the Zariski closure of polynomials with symmetric rank r. It bounds and is closely related to rank. This dissertation demonstrates new lower bounds for the symmetric border rank of the polynomial detn. We prove this result using methods from algebraic geometry and representation theory. In addition to the lower bounds for symmetric border rank of detn, we present a lower bound for symmetric border rank of a related polynomial, perm3. We conclude by giving future directions for continuing this project. The first direction is to use the methods from algebraic geometry and representation theory used in this dissertation to study permn. With the new lower bound on symmetric border rank of perm3 we know that there are only 3 possible values for symmetric border rank of perm3. One could ask which of the 3 possible values for symmetric border rank of perm3 is the correct value.