Decompositions of powers of quadrics

TL;DR

This work addresses the Waring decomposition problem for powers of a quadratic form $q_n^s$ over $\mathbb{C}$, aiming to determine rank and construct near-minimal decompositions. It develops an apolarity framework aligned with harmonic analysis, proving the apolar ideal of $q_n^s$ is generated by all degree $s+1$ harmonic polynomials, and uses the harmonic decomposition $S^d\mathbb{C}^n=\bigoplus_{j} q_n^j\mathcal{H}_{n,d-2j}$ to control catalecticant maps. Building on Reznick’s real-decomposition results, the paper extends to the complex setting, establishing tight decompositions in several regimes, and provides exact rank results for $q_n^2$ in many variables, including a complete treatment in two variables. It connects Waring ranks with spherical designs and orthogonal-invariant theory, yielding new decomposition families and growth bounds for ranks as $n$ grows. Overall, the results enhance the understanding of ranks and explicit decompositions for powers of quadrics, with implications for invariant theory and algebraic geometry of symmetric tensors.

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.

Figures (8)

• Figure 4.1: Examplesofpointsofdecompositionsoftheforms$q_2^3$and$q_2^4$.Theblueoctagonontheleftgivesthe$4$projectivepointsobtainedasrootsofthepolynomial$u^4-v^4$,whilethereddecagonontherightgivesthe$5$projectivepointsobtainedasrootsofthepolynomial$u^5-v^5$(pointsoppositetotheoriginrepresentthesamepointin$\mathbb{P}^1(\mathbb{C})$).
• Figure 4.2: Graphicalrepresentationofdecomposition\ref{['rel_decomp_icos']},whoseelementscorrespond,uptocentralsimmetry,totheverticesofaregularicosahedron.
• Figure 5.1: Graphicalrepresentationofdecomposition\ref{['decom_n=3_s=4_std']},whoseelementscorrespond,uptocentralsimmetry,totheverticesofaregularicosahedron(inblue)togheterwiththeverticesofaregulardodecahedron(inorange).
• Figure 5.2: Diagramrepresentingtheweightsofthespaceofharmonicpolynomialsdefinedbyformulas\ref{['rel_basis_weights_harmonic_polynomials']}.
• Figure 5.3: Graphicalrepresentationofdecomposition\ref{['rel_decom_n=3_s=2_uno']}instandardcoordinates$\{y_1,y_2,y_3\}$,whoseelementscorrespond,uptocentralsimmetry,totheverticesoftwotriangles,respectivelyinredandblue,placedatdifferentheights.

Theorems & Definitions (156)

• Theorem : Rez92*Proposition9.2
• Theorem
• Theorem
• Theorem
• Theorem
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 1
• Definition 2.4