Border rank of powers of ternary quadratic forms

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 qn=x12+…+xn2 in an arbitrary number n of variables. We determine the apolar ideal of any power qns, 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 q3. After verifying certain properties, we utilize the recent technique of border apolarity to establish that the border rank of any power q3s is equal to the rank of its middle catalecticant matrix, namely (s+1)(s+2)/2.