Decompositions of powers of quadrics

TL;DR

This paper studies the Waring decompositions of powers of quadratic forms over complex numbers, providing new insights into their ranks, minimal decompositions, and the structure of their apolar ideals, with extensions to real forms.

Contribution

It offers a detailed method for analyzing the apolar ideal of quadratic powers, generalizes previous results on real decompositions, and investigates ranks of specific quadratic powers.

Findings

Apolar ideal of s-th power of quadratic form generated by harmonic polynomials of degree s+1

New minimal decompositions for real and complex quadratic powers

Rank of second power of non-degenerate quadratic forms in n variables is (n^2 + n + 2)/2

Abstract

We analyze the problem of determining Waring decompositions of the powers of any quadratic form over the field of complex numbers. Our main goal is to provide information about their rank and also to obtain decompositions whose size is as close as possible to this value. This is a classical problem and these forms assume importance especially because of their invariance under the action of the special orthogonal group. We give the detailed procedure to prove that the apolar ideal of the $s$-th power of a quadratic form is generated by the harmonic polynomials of degree $s+1$. We also generalize and improve some of the results on real decompositions given by B. Reznick in his notes of 1992, focusing on possibly minimal decompositions and providing new ones, both real and complex. We investigate the rank of the second power of a non-degenerate quadratic form in $n$ variables, which in most cases is equal to $(n^2+n+2)/2$, and also give some results on powers of ternary quadratic forms.