[Paper Review] The Waring rank of the sum of pairwise coprime monomials

Enrico Carlini, Maria Virginia Catalisano+1 more|arXiv (Cornell University)|Dec 15, 2011

Graph Labeling and Dimension Problems6 references3 citations

TL;DR

This paper determines the Waring rank of any polynomial that is a sum of pairwise coprime monomials, proving that the rank is additive across such monomials. It provides a complete formula for the Waring rank of any monomial and constructs minimal sum-of-powers decompositions, showing that certain monomials in three variables can have higher rank than generic forms of the same degree.

ABSTRACT

In this paper we compute the Waring rank of any polynomial of the form F=M_1+...+M_r, where the M_i are pairwise coprime monomials, i.e., GCD(M_i,M_j)=1 for i not j. In particular, we determine the Waring rank of any monomial. As an application we show that certain monomials in three variables give examples of forms of rank higher than the generic form. As a further application we produce a sum of power decomposition for any form which is the sum of pairwise coprime monomials.

Motivation & Objective

• To determine the Waring rank of any polynomial that is a sum of pairwise coprime monomials.
• To compute the Waring rank of any individual monomial in n variables.
• To construct minimal sum-of-powers decompositions for such polynomials.
• To identify cases where monomials have higher Waring rank than generic forms of the same degree.
• To establish that the Waring rank is independent of the ambient polynomial ring's variable count.

Proposed method

• Uses the Apolarity Lemma to relate Waring rank to the ideal of reduced points contained in the annihilator ideal $F^\perp$.
• Applies the Hilbert function and multiplicity theory to analyze one-dimensional saturated ideals of points.
• Characterizes the Waring rank of a monomial $x_1^{a_1} \cdots x_n^{a_n}$ as $\prod_{i=2}^n (a_i + 1)$ when $1 \leq a_1 \leq \cdots \leq a_n$.
• Constructs a minimal sum-of-powers decomposition by selecting roots of unity to define linear forms.
• Uses the fact that adding variables does not reduce the Waring rank, ensuring invariance across polynomial rings.
• Employs the structure of the annihilator ideal $M^\perp$ to identify the set of points corresponding to the decomposition.

Experimental results

Research questions

• RQ1 What is the Waring rank of a sum of pairwise coprime monomials in $n$ variables?
• RQ2 Can the Waring rank of a monomial be computed in closed form for any degree and variable distribution?
• RQ3 Do any monomials in three variables have Waring rank strictly greater than that of a generic form of the same degree?
• RQ4 Is the Waring rank of a polynomial invariant under the addition of variables to the polynomial ring?
• RQ5 Can a minimal sum-of-powers decomposition be explicitly constructed for sums of coprime monomials?

Key findings

• The Waring rank of a monomial $x_1^{a_1} \cdots x_n^{a_n}$ with $1 \leq a_1 \leq \cdots \leq a_n$ is $\prod_{i=2}^n (a_i + 1)$ for $n \geq 2$, and 1 for $n = 1$.
• For the sum $F = \sum_{i=1}^r M_i$ of pairwise coprime monomials, the Waring rank satisfies $\mathrm{rk}(F) = \sum_{i=1}^r \mathrm{rk}(M_i)$.
• In three variables, certain monomials such as $x_1 x_2^{(d-1)/2} x_3^{(d-1)/2}$ (for odd $d$) achieve a maximal rank asymptotically equal to $d^2/4$, exceeding the generic form rank of approximately $d^2/6$.
• For $n \geq 4$, no monomial can have Waring rank exceeding that of a generic form of the same degree, as the ratio of maximal monomial rank to generic rank is at most 1.
• A minimal sum-of-powers decomposition for any sum of pairwise coprime monomials is obtained by independently decomposing each monomial using roots of unity, with the total number of terms equal to the sum of individual ranks.
• The decomposition for a monomial $M = x_1^{a_1} \cdots x_n^{a_n}$ uses linear forms $x_1 + \epsilon(2)x_2 + \cdots + \epsilon(n)x_n$ where $\epsilon(i)$ ranges over the $(a_i+1)$-th roots of unity, and the coefficients $\gamma_{\epsilon(2),\ldots,\epsilon(n)}$ are determined by solving a linear system.