Please note: This master’s thesis presentation will take place in DC 2314.
Abhiroop Sanyal, Master’s candidate
David R. Cheriton School of Computer Science
Supervisor: Professor Rafael Oliveira
We study the problem of decomposition of homogeneous polynomials into sums of powers of linear forms (or limits of sums of powers of linear forms) with minimal number of summands, also known as the Waring rank (respectively, border Waring rank) of the polynomial. The case for generic polynomials was resolved by Alexander and Hirschowitz in 1995. However, determining the Waring rank (respectively, border Waring rank) of a given polynomial is computationally NP-hard and finding rank of many explicit classes of polynomials remain interesting open problems.
A related interesting problem is to determine if a given homogeneous polynomial can be written as the sum of two homogeneous polynomials in independent sets of variables, after a linear change in coordinates (studied extensively by Buczyńska et al and Kleppe). Such polynomials are said to be direct sums. A central tool for studying both these problems is apolarity theory, which is the study of the ideal of annihilators of a given polynomial.
In this thesis, we focus our attention on a very special class of polynomials, called 1-support concentrated polynomials: whose space of partial derivatives is spanned by partial derivatives with respect to powers of independent linear forms. We show that this class of polynomials is the same as the class of polynomials whose dimension of partial derivatives, at each degree, is at most the number of variables. We show that polynomials of this class of sufficiently high degree can be written as a limit of direct sums. In the low-degree regime, we focus on the generalization of a class of polynomials introduced by Stanley and show that these polynomials can also be written as limits of direct sums. We also prove a lower bound on the Waring rank of Stanley polynomials and show a separation between Waring rank and border Waring rank of bivariate Stanley polynomials of all degree as well as in the case of some smaller examples of Stanley polynomials.