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.

翻译：我们研究多线性映射奇异性的计算复杂性。尽管行列式刻画了矩阵的奇异性，但其多线性推广——超行列式——仅在边界格式中有定义，且很快变得代数上难以处理。我们证明张量奇异性的内在概念，即退化性，对实数存在性理论是完备的。该归约是精确且完全代数的：齐次二次可行性被归约为射影双线性可行性，进而归约为奇异矩阵束可行性，最后直接编码为张量退化性。未使用任何组合工具。在边界格式中，退化性与超行列式为零相重合。因此我们分离了内在张量奇异性与其经典多项式证书之间的确切差距。我们证明确定性困难性向超行列式的转移可归约为选择完成多项式零点集之外的一点，从而得到多项式恒等测试的一个结构化实例。我们进一步形式化了几种自然确定性嵌入策略的失败。这确立了一个尖锐的边界：实3-张量退化性在\(\ER\)-完全性层次上被完全刻画，而超行列式为零的确定性复杂性仍与代数复杂性中的去随机化问题紧密相关。