Waring rank, Border rank and support concentration of partials

Abstract

In this thesis, we study the classical problem of decomposition of a homogeneous polynomial into sums of powers of linear forms with minimal summands (also known as the Waring Rank of the polynomial) and related problems. The case of quadratic polynomials and binary forms was studied by Sylvester in his seminal paper in 1851 and the case for generic polynomials was resolved more than a century later by Alexander and Hirschowitz in 1995. The problem is NP-hard computationally and finding the Waring rank for several interesting classes of polynomials, for example, the general n*n symbolic determinant/permanent, remains an open problem.

An important parameter in the study of this problem is the dimension D of the vector space of partial derivatives of the given polynomial. It is known that if the Waring rank of a polynomial in n variables of degree d is s, then D is at most s*(d+1). A longstanding conjecture states that given D, the Waring rank is upper bounded by a polynomial in n, d, and D.

To study this conjecture, we restrict to a very special class of polynomials with no redundant variables (also called concise), which we call 1-support concentrated polynomials, that are defined by the following property: Given such a polynomial f, all its partial derivatives can be obtained as linear combinations of derivatives with respect to powers of a fixed set of n linearly independent linear forms. A crucial property of such f is that the dimension of partial derivatives of f at any degree is at most n. We show that the converse is true: any concise polynomial for which dimension of partial derivatives at any degree is less than n, is also 1-support concentrated. We also generalize an example given by Stanley to give an explicit class of concise polynomials ST(n,d) in O(max{n^d, d^n}) variables of degree d that is 1-support concentrated.

A polynomial is a Direct Sum if it can be written as a sum of two polynomials in distinct sets of variables, up to a linear change of variables. A polynomial f is a limit of direct sums if there is a sequence of polynomials, each a direct sum, converging to f. Necessary and sufficient conditions for a polynomial to be a direct sum or a limit of direct sums were extensively studied by Buczy\'nska et al. and Kleppe. We show that any concise 1-support concentrated polynomial in n variables with degree d > 2n is a limit of direct sums. We also show that ST(n,d) (which does not satisfy the previous degree hypothesis) is a limit of direct sums.

The border rank of a homogeneous polynomial f is the minimal r such that there is a sequence of polynomials, each with Waring rank at most r, converging to f. The debordering question is as follows: given f with border Waring rank r, what is the best upper bound for Waring rank of f in terms of n,r, and d? The best-known bound is due to Dutta et al., in 2024. In the context of this problem, it is interesting to find examples f for which the Waring rank of f is strictly greater than its border Waring rank. We show that ST(3,4) and ST(2,d), for any d > 2, have this property.