[Paper Review] Symmetric tensors: rank, Strassen's conjecture and e-computability

Enrico Carlini, Maria Virginia Catalisano+3 more|arXiv (Cornell University)|Jun 10, 2015

Tensor decomposition and applications7 references4 citations

TL;DR

This paper introduces the concept of $e$-computability to establish new lower bounds for the Waring rank of symmetric tensors and proves Strassen's Conjecture holds in infinitely many new cases. By linking the rank of a form to the Hilbert function of its apolar ideal and using geometric constraints on zero-dimensional schemes, the authors derive precise rank computations and validate the conjecture for several infinite families of forms.

ABSTRACT

In this paper we introduce a new method to produce lower bounds for the Waring rank of symmetric tensors. We also introduce the notion of $e$-computability and we use it to prove that Strassen's Conjecture holds in infinitely many new cases.

Motivation & Objective

• To develop a new method for computing lower bounds on the Waring rank of symmetric tensors.
• To introduce and study the notion of $e$-computability as a tool for rank computation.
• To prove Strassen's Conjecture—additivity of rank across disjoint variable sets—holds for infinitely many new families of forms.
• To provide exact rank computations for specific symmetric forms using geometric and algebraic constraints on apolar ideals.
• To demonstrate that certain forms are not 1-computable, showing limitations of existing computational techniques.

Proposed method

• Define $e$-computability as a condition on the apolar ideal $F^{ot}$ and a linear subspace $I$ such that the rank of $F$ equals the sum of the Hilbert function of $T/(F^{ot}:I + (t))$ for a general linear form $t$.
• Use the Apolarity Lemma to relate Waring decompositions to zero-dimensional schemes apolar to $F$, where $I_{ ext{scheme}} \subset F^{ot}$.
• Apply Hilbert function analysis to derive lower bounds on rank via $\sum_i HF(T/\tilde{I}, i)$, where $\tilde{I} = F^{ot}:(X_i) + (X_i)$.
• Use geometric constraints: if $X_iX_j + c_{ij}X_0^2 \in I_{\mathbb{X}}$, then $c_{ij} = 0$ unless $a=1$, to eliminate spurious solutions.
• Leverage the structure of $F^{ot}$ for specific forms (e.g., $F = x_0^a(x_1^b + \cdots + x_n^b)$) to compute $\mathrm{rk}(F)$ via ideal containment and Hilbert function summation.
• Use contradiction arguments based on point counts on lines and the union of lines to rule out minimal decompositions of certain sizes.

Experimental results

Research questions

• RQ1 For which symmetric forms can the Waring rank be computed using $e$-computability?
• RQ2 Does Strassen's Conjecture—that rank is additive over disjoint variable sets—hold for new families of forms?
• RQ3 Can $e$-computability be used to establish sharp lower bounds on the Waring rank?
• RQ4 Are there forms for which standard computational techniques (e.g., 1-computability) fail, and if so, how can their rank still be computed?
• RQ5 What geometric and algebraic conditions on apolar ideals ensure that a given decomposition is minimal?

Key findings

• The form $F = x_0^a(x_1^b + \cdots + x_n^b)$ with $2 \leq a+1 \leq b$ and $n \geq 3$ has rank between $bn - n + 3$ and $bn$, and for $n=3$, $\mathrm{rk}(F) = 3b$.
• The form $F = w(x^3 + y^3 + z^3)$ in $k[x,y,z,w]$ has rank 9 and is not 1-computable, as shown by contradiction in Hilbert function sums.
• Strassen's Conjecture holds for infinitely many new families of forms, including monomials and the $x_0^a(x_1^b + \cdots + x_n^b)$ class.
• The rank of $F = x_0^a(x_1^b + \cdots + x_n^b)$ is exactly $bn$ when $a \geq 1$, and the lower bound $bn - n + 3$ is sharp.
• For $F = x_0^a(x_1^b + \cdots + x_n^b)$, the ideal $F^{ot}$ contains $X_0^{a+1}$ and relations $X_iX_j$ and $X_1^b - X_i^b$, which constrain the apolar schemes.
• The proof that $\mathrm{rk}(F) = bn - n + 3$ relies on showing that $\sum_i HF(T/\tilde{I}, i) = bn - n + 2$ and that $\mathrm{rk}(F) > bn - n + 2$ via contradiction on point distribution over lines.