Border rank of powers of ternary quadratic forms

TL;DR

This paper determines the border rank of powers of quadratic forms in three variables, revealing that it equals the rank of the middle catalecticant matrix, and introduces new algebraic techniques for this calculation.

Contribution

It introduces a method to compute the border rank of powers of quadratic forms in three variables using border apolarity and catalecticant matrices, extending previous results.

Findings

Border rank of powers of quadratic forms in three variables equals the rank of the middle catalecticant matrix.

Apolar ideal of powers of quadratic forms is generated by harmonic polynomials.

The technique applies to non-degenerate quadratic forms, generalizing prior work.

Abstract

We determine the border rank of each power of any quadratic form in three variables. Since the problem for rank $1$ and rank $2$ quadratic forms can be reduced to determining the rank of powers of binary forms, we primarily focus on non-degenerate quadratic forms. We begin by considering the quadratic form $q_{n}=x_1^{2}+\dots+x_n^{2}$ in an arbitrary number $n$ of variables. We determine the apolar ideal of any power $q_n^s$, proving that it corresponds to the homogeneous ideal generated by the harmonic polynomials of degree $s+1$. Using this result, we select a specific ideal contained in the apolar ideal for each power of a quadratic form in three variables, which, without loss of generality, we assume to be the form $q_3$. After verifying certain properties, we utilize the recent technique of border apolarity to establish that the border rank of any power $q_3^s$ is equal to the rank of its middle catalecticant matrix, namely $(s+1)(s+2)/2$.