[Paper Review] Symmetric tensors: rank and Strassen's conjecture

Enrico Carlini, Maria Virginia Catalisano+3 more|arXiv (Cornell University)|Dec 9, 2014

Tensor decomposition and applications9 references4 citations

TL;DR

This paper introduces the concept of linear computability to determine the Waring rank of symmetric tensors, offering a novel method to identify forms that satisfy Strassen's Conjecture. It establishes infinitely many new examples where the conjecture holds, advancing the understanding of tensor rank and algebraic complexity.

ABSTRACT

In this paper we introduce the notion of linear computability as a method of finding the Waring rank of forms. We use this notion to find infinitely many new examples which satisfy Strassen's Conjecture.

Motivation & Objective

• To develop a new method—linear computability—for analyzing the Waring rank of symmetric forms.
• To address Strassen's Conjecture, which posits a lower bound on the rank of certain tensors.
• To identify and construct infinitely many new cases where Strassen's Conjecture is satisfied.
• To extend the known class of forms for which the Waring rank can be precisely determined.

Proposed method

• Introduces the notion of linear computability as a structural property of symmetric tensors.
• Applies linear computability to derive conditions under which the Waring rank of a form can be computed.
• Uses algebraic geometry and polynomial identities to characterize forms with specific rank behavior.
• Leverages the structure of symmetric tensors to identify invariants that imply rank minimality.
• Establishes a criterion based on linear dependence relations among derivatives to detect computability.
• Applies the criterion to construct explicit families of forms satisfying Strassen's Conjecture.

Experimental results

Research questions

• RQ1 Can linear computability be used to determine the Waring rank of symmetric tensors?
• RQ2 Which symmetric forms satisfy Strassen's Conjecture, and how can they be systematically constructed?
• RQ3 What algebraic conditions imply that a symmetric form has minimal Waring rank?
• RQ4 How does linear computability relate to the rank of a symmetric tensor?
• RQ5 Are there infinite families of forms for which Strassen's Conjecture holds?

Key findings

• The paper proves that linear computability implies a bound on the Waring rank, enabling exact rank computation.
• It identifies infinitely many new families of symmetric forms that satisfy Strassen's Conjecture.
• The method provides a constructive framework to verify the conjecture for specific classes of forms.
• Linear computability is shown to be a sufficient condition for rank minimality in certain symmetric tensor settings.
• The approach offers a systematic algebraic criterion to detect forms with minimal Waring rank.
• The results extend the known range of cases where Strassen's Conjecture is valid, without relying on case-by-case analysis.