Waring decompositions of the product of two quadrics: the small rank cases

TL;DR

This paper investigates the Waring decompositions of products of two quadrics, providing detailed descriptions for small cases and new bounds for larger ones, advancing understanding of their algebraic structure.

Contribution

It offers a comprehensive analysis of minimal apolar sets for specific cases and establishes new bounds for Waring rank when both parameters are at least 3.

Findings

Described forbidden locus, structure, and Hilbert function for small cases.

Proved all minimal apolar ideals share the same Hilbert function.

Established new lower and upper bounds for Waring rank for larger cases.

Abstract

In this paper we study forms of the type $(x_1^2+ \cdots +x_m^2)(y_1^2+ \cdots+y_n^2)$ using projections. For $m=1, m=2$, and for any $n$ we describe: the forbidden locus, the structure and the Hilbert function of all minimal apolar sets. In particular, we show that every minimal apolar ideal has the same Hilbert function. For $m,n \geq 3,$ we provide new lower and upper bounds for the Waring rank.