Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Meghana Bhat, Enrico Carlini, Saipriya Dubey, Shreedevi K. Masuti

TL;DR

This work investigates Waring decompositions for products of two quadrics through apolarity and projection techniques, focusing on small-rank cases and providing both exact results and bounds. For the single-variable product F = x^2(y_1^2+...+y_n^2), the authors prove the rank is 3n and determine a precise structure for all minimal apolar sets, including the projection geometry and the forbidden locus. For the two-quadrics product F = (x_1^2+x_2^2)(y_1^2+...+y_n^2), they establish a rank of 4n and describe the apolar set structure via two quadrics in the ideal, along with a detailed analysis of projections and Hilbert functions. The paper then generalizes to F = (x_1^2+...+x_m^2)(y_1^2+...+y_n^2), deriving bounds Rk(F) ∈ [n(m+2), 2mn], and extends to the more general form F = x_1^{a_1}...x_m^{a_m}(y_1^b+...+y_n^b) with Rk(F) = n∏(a_i+1) under specified conditions, supported by explicit apolar constructions and e-computability arguments. These results illuminate the geometry of minimal apolar sets and provide a framework for approaching rank questions in broader reducible forms.

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.

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

TL;DR

This work investigates Waring decompositions for products of two quadrics through apolarity and projection techniques, focusing on small-rank cases and providing both exact results and bounds. For the single-variable product F = x^2(y_1^2+...+y_n^2), the authors prove the rank is 3n and determine a precise structure for all minimal apolar sets, including the projection geometry and the forbidden locus. For the two-quadrics product F = (x_1^2+x_2^2)(y_1^2+...+y_n^2), they establish a rank of 4n and describe the apolar set structure via two quadrics in the ideal, along with a detailed analysis of projections and Hilbert functions. The paper then generalizes to F = (x_1^2+...+x_m^2)(y_1^2+...+y_n^2), deriving bounds Rk(F) ∈ [n(m+2), 2mn], and extends to the more general form F = x_1^{a_1}...x_m^{a_m}(y_1^b+...+y_n^b) with Rk(F) = n∏(a_i+1) under specified conditions, supported by explicit apolar constructions and e-computability arguments. These results illuminate the geometry of minimal apolar sets and provide a framework for approaching rank questions in broader reducible forms.

Abstract

In this paper we study forms of the type using projections. For , and for any 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 we provide new lower and upper bounds for the Waring rank.

Paper Structure

This paper contains 6 sections, 21 theorems, 124 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background material
  3. Basic results
  4. The form $x^2(y_1^2+\cdots+y_n^2)$
  5. The form $(x_1^2+x_2^2)(y_1^2+\cdots+y_n^2)$
  6. The forms $(x_1^2+x_2^2+ \cdots+x_m^2)(y_1^2+y_2^2+ \cdots+y_n^2)$

Key Result

Lemma 2.1

Let $\mathbb{X}=\{P_1, \ldots, P_r\} \subset \mathbb{P}^n$ be a set of points. Then for some $\alpha_i \in \mathbb{C}$, we have $F = \sum_{i=1}^r\alpha_i L_i^d$ if and only if $I(\mathbb{X}) \subset F^{\perp},$ where $L_i$ is the linear form corresponding the point $P_i$ for $1 \leq i \leq r.$

Figures (4)

  • Figure 1: Structure of a minimal apolar set of $F=x^2(y_1^2+\cdots+y_n^2)$: red dots represent apolar points contained in $n$-lines with 3 points on each line.
  • Figure 2: Structure of a minimal apolar set of $F=(x_1^2+x_2^2)(y_1^2+\cdots+y_n^2)$: black dots represent minimal apolar points contained in $n$-planes with 4 points on each plane.
  • Figure 3: Minimal apolar set whose projection from $V(X_1,X_2)$ to $V(Y_1,Y_2,Y_3)$ having 2 points.
  • Figure 4: Minimal apolar set whose projection from $V(X_1,X_2)$ to $V(Y_1,Y_2,Y_3)$ having 6 points.

Theorems & Definitions (51)

  • Example 1.3
  • Lemma 2.1: Apolarity Lemma
  • Theorem 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 41 more