[Paper Review] Spaces of Sums of Powers and Real Rank Boundaries

Mateusz Michałek, Hyunsuk Moon|arXiv (Cornell University)|Dec 23, 2016

Algebraic Geometry and Number Theory16 references4 citations

TL;DR

This paper investigates real Waring decompositions of homogeneous polynomials, focusing on the algebraic boundaries of spaces of sums of powers (SSP) and real rank boundaries. For quaternary quadrics, it explicitly describes the algebraic boundary of SSP using anti-polar forms and relates it to loci in the Hilbert scheme. The key result is that the real rank boundary for quaternary cubics is a degree-40 hypersurface, identified as the join of the third secant variety and tangential variety of the third Veronese embedding of ℙ³.

ABSTRACT

We investigate properties of Waring decompositions of real homogeneous forms. We study the moduli of real decompositions, so-called Space of Sums of Powers, naturally included in the Variety of Sums of Powers. Explicit results are obtained for quaternary quadrics, relating the algebraic boundary of ${ m SSP}$ to various loci in the Hilbert scheme of four points in $\mathbb{P}^3$. Further, we study the locus of general real forms whose real rank coincides with the complex rank. In case of quaternary quadrics the boundary of this locus is a degree forty hypersurface $J(σ_3(v_3(\mathbb{P}^3)),τ(v_3(\mathbb{P}^3)))$.

Motivation & Objective

• To understand the geometry and topology of real Waring decompositions, particularly the moduli space of real decompositions (SSP) within the variety of sums of powers (VSP).
• To characterize the algebraic boundary of SSP(f) for real homogeneous forms, especially in the case of quaternary quadrics.
• To determine when a real form admits a real Waring decomposition with rank equal to its complex rank, and to describe the algebraic boundary of this locus.
• To develop and implement a fast, deterministic algorithm for computing Waring decompositions of general quaternary cubics.
• To relate the real rank boundary to classical algebraic geometry objects such as secant varieties, tangential varieties, and dual varieties of symmetroids.

Proposed method

• Introduces the anti-polar form Ω(f) for even-degree homogeneous forms, generalizing the dual quadric, to analyze apolar schemes and nonreduced structures.
• Uses the middle catalecticant map Af: S^d(V) → S^d(V*) to define and study the anti-polar form, enabling explicit computation of algebraic boundaries.
• Applies elimination theory and Gröbner basis techniques to compute the algebraic boundary of SSP(f) by eliminating variables from systems of equations derived from apolarity and flattening relations.
• Employs computational algebra systems (e.g., Macaulay2) to verify irreducibility, compute real loci, and analyze resolutions of ideals over ℂ and ℝ.
• Utilizes the join construction J(σ₃(v₃(ℙ³)), τ(v₃(ℙ³))) to describe the real rank boundary for quaternary cubics.
• Applies primary decomposition and singular locus analysis to study the geometry of the boundary and its components, particularly in the quinary quadric case.

Experimental results

Research questions

• RQ1 What is the algebraic boundary of the space of sums of powers (SSP) for a real homogeneous form, particularly for quaternary quadrics?
• RQ2 How does the real rank of a form relate to its complex rank, and what is the algebraic boundary of the locus where they coincide?
• RQ3 Can a deterministic, fast algorithm be constructed to compute the unique Waring decomposition of a general quaternary cubic?
• RQ4 How is the real rank boundary for quaternary cubics related to classical algebraic varieties such as secant and tangential varieties?
• RQ5 What is the geometric and algebraic structure of the boundary of SSP(f) in relation to the Hilbert scheme of points?

Key findings

• For quaternary quadrics, the algebraic boundary of SSP(f) is explicitly described using the anti-polar form and is related to loci in the Hilbert scheme of four points in ℙ³.
• The real rank boundary ∂ₐₗg(ℛ₄,₃) for quaternary cubics is an irreducible hypersurface of degree 40 in ℙ¹⁹, identified as the join variety J(σ₃(v₃(ℙ³)), τ(v₃(ℙ³))).
• A fast, deterministic algorithm is implemented that computes the unique Waring decomposition of a general quaternary cubic, applicable both numerically and parametrically over K(t).
• One component of the real rank boundary for quaternary quartics is shown to be the dual of the variety of quartic symmetroids.
• The algebraic boundary of SSP(f) for quinary quadrics is computed via elimination and Gröbner basis methods, yielding a degree 40 hypersurface.
• The real locus of the boundary variety is analyzed via primary decomposition and singular locus computation, revealing that the singular locus lies on a union of irreducible components defined by quadratic equations.