$\exists\mathbb{R}$-Completeness of Tensor Degeneracy and a Derandomization Barrier for Hyperdeterminants

2026-04-18

AI Summary

This work investigates the computational complexity of singularities in multilinear maps, focusing on the fundamental distinction between tensor degeneracy and the vanishing of hyperdeterminants. By means of a purely algebraic reduction, the authors transform the homogeneous quadratic feasibility problem into the problem of determining degeneracy of third-order real tensors, thereby establishing—for the first time—that the latter is complete for the existential theory of the reals ($\exists \mathbb{R}$). Their approach integrates techniques from algebraic complexity theory, including projective bilinear feasibility analysis, singular matrix pencil encodings, and careful selection of polynomial zero sets. The main contributions are the proof of $\exists \mathbb{R}$-completeness for tensor degeneracy and the identification of structured polynomial identity testing as the core obstacle to deterministic hyperdeterminant computation, formally demonstrating the failure of several natural derandomization strategies.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

We study the computational complexity of singularity for multilinear maps. While the determinant characterizes singularity for matrices, its multilinear analogue -- the hyperdeterminant -- is defined only in boundary format and quickly becomes algebraically unwieldy. We show that the intrinsic notion of tensor singularity, namely degeneracy, is complete for the existential theory of the reals. The reduction is exact and entirely algebraic: homogeneous quadratic feasibility is reduced to projective bilinear feasibility, then to singular matrix-pencil feasibility, and finally encoded directly as tensor degeneracy. No combinatorial gadgets are used. In boundary format, degeneracy coincides with hyperdeterminant vanishing. We therefore isolate the exact gap between intrinsic tensor singularity and its classical polynomial certificate. We show that deterministic hardness transfer to the hyperdeterminant reduces to selecting a point outside the zero set of a completion polynomial, yielding a structured instance of polynomial identity testing. We further formalize the failure of several natural deterministic embedding strategies. This identifies a sharp frontier: real 3-tensor degeneracy is fully characterized at the level of \(\ER\)-completeness, while the deterministic complexity of hyperdeterminant vanishing remains tied to a derandomization problem in algebraic complexity.

Problem

• tensor degeneracy
• hyperdeterminant
• existential theory of the reals
• computational complexity
• derandomization

Innovation

• tensor degeneracy
• existential theory of the reals
• hyperdeterminant
• polynomial identity testing
• algebraic complexity