[Paper Review] On the typical rank of real bivariate polynomials

Edoardo Ballico|arXiv (Cornell University)|Apr 14, 2012

Tensor decomposition and applications9 references4 citations

TL;DR

This paper establishes that for real bivariate homogeneous polynomials of degree $d \geq 6$, the ranks $d-1$ and, when $d$ is odd, $(d+3)/2$ are typical ranks—meaning they occur generically in an open Euclidean subset of the real coefficient space. The results confirm parts of Comon and Ottaviani's conjecture on typical ranks for real bivariate forms, extending prior work on $d \leq 5$ and proving the existence of multiple typical ranks for $d \geq 6$. The proof relies on algebraic geometry techniques, including Veronese embeddings, real and complex linear spans, and projection arguments on rational normal curves.

ABSTRACT

Here we study the typical rank for real bivariate homogeneous polynomials of degree $d\ge 6$ (the case $d\le 5$ being settled by P. Comon and G. Ottaviani). We prove that $d-1$ is a typical rank and that if $d$ is odd, then $(d+3)/2$ is a typical rank.

Motivation & Objective

• To determine the typical ranks of real bivariate homogeneous polynomials of degree $d \geq 6$, extending prior results for $d \leq 5$.
• To verify the Comon-Ottaviani conjecture that all integers $t$ satisfying $\lfloor(d+2)/2\rfloor \leq t \leq d$ are typical ranks for real bivariate forms.
• To establish the existence of multiple typical ranks for $d \geq 6$, particularly showing $d-1$ is typical and $(d+3)/2$ is typical when $d$ is odd.
• To analyze the real rank via geometric methods, including real and complex linear spans on the Veronese variety and projection from points on the rational normal curve.

Proposed method

• The study uses the Veronese embedding $\nu_d: \mathbb{P}^1(\mathbb{R}) \to \mathbb{P}^d(\mathbb{R})$ to represent homogeneous polynomials as points on the rational normal curve $C_d(\mathbb{R})$.
• The real rank $Rsr(f)$ is defined as the minimal number of real points on $C_d(\mathbb{R})$ whose real linear span contains the point $P$ corresponding to $f$.
• The proof employs linear projection $\ell_Q: \mathbb{P}^d(\mathbb{R}) \setminus \{Q\} \to \mathbb{P}^{d-1}(\mathbb{R})$ from a point $Q = \nu_d(O)$ associated to a linear form $R$, reducing the problem to lower-degree polynomials.
• A contradiction argument is used: assuming $Rsr(f) \leq d-2$ leads to a configuration of points on the projected space violating Lemma 1, which bounds the size of sets with non-unique linear spans.
• The existence of a non-empty Euclidean open set of polynomials with real rank $d-1$ is shown by constructing a family $g_{f,c,R} = \sum c_i L_i^d + cR^d$ with $f$ of degree $d-1$ and $f$ having $d-1$ distinct real roots.
• The construction ensures that for generic $f$ and $c$, the resulting polynomial $g_{f,c,R}$ has distinct roots and not all real, confirming the openness of the set of such polynomials in the Euclidean topology.

Experimental results

Research questions

• RQ1 Is $d-1$ a typical rank for real bivariate homogeneous polynomials of degree $d \geq 6$?
• RQ2 For odd $d \geq 5$, is $(d+3)/2$ a typical rank in the real bivariate case?
• RQ3 Does the Comon-Ottaviani conjecture—that all integers $t$ with $\lfloor(d+2)/2\rfloor \leq t \leq d$ are typical ranks—hold for $d \geq 6$?
• RQ4 Can the real rank of a bivariate form be characterized geometrically via linear spans on the Veronese variety over $\mathbb{R}$ and $\mathbb{C}$?
• RQ5 What is the minimal number of typical ranks for real bivariate forms of degree $d \geq 6$?

Key findings

• For all $d \geq 6$, the integer $d-1$ is a typical rank for real bivariate homogeneous polynomials of degree $d$, meaning there exists a non-empty open Euclidean subset of $\mathbb{R}[x,y]_d$ where all polynomials have real rank $d-1$.
• For odd $d = 2m+1 \geq 5$, the integers $m+1$, $m+2$, and $2m+1 = d$ are all typical ranks, confirming the existence of at least four typical ranks when $d \geq 7$.
• The paper proves that $d-1$ is a typical rank by constructing a family of polynomials $g_{f,c,R} = \sum c_i L_i^d + cR^d$ with $f$ of degree $d-1$ and $d-1$ distinct real roots, showing that such polynomials have real rank $d-1$ generically.
• The construction ensures that for generic $f$ and $c$, the polynomial $g_{f,c,R}$ has distinct roots and not all real, so the set of such polynomials forms a non-empty open subset in the Euclidean topology.
• The proof relies on contradiction: assuming $Rsr(g_{f,c,R}) \leq d-2$ leads to a contradiction via Lemma 1, which bounds the size of sets of points on the rational normal curve with non-unique linear spans.
• The result confirms that $d-1$ is a typical rank and, combined with known results, supports the Comon-Ottaviani conjecture for $d \leq 7$, showing at least three typical ranks for even $d \geq 6$ and four for odd $d \geq 7$.