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

TL;DR

This paper proves that tensor degeneracy is -complete, establishing a precise complexity classification and highlighting a derandomization barrier for hyperdeterminants in algebraic complexity theory.

Contribution

It establishes the -completeness of tensor degeneracy and clarifies the complexity gap between tensor singularity and hyperdeterminant computation.

Findings

Tensor degeneracy is -complete.

Hyperdeterminant vanishing complexity is linked to derandomization.

Natural deterministic embedding strategies fail for hyperdeterminants.

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.