We study the decision version of tensor spectral norm from the viewpoint of real algebraic complexity. For a rationally specified tensor, the tensor spectral threshold problem asks whether its spectral norm exceeds a prescribed rational threshold. Since the feasible domain is compact, attainment itself is trivial; the meaningful question is the threshold decision problem. We prove that this problem is $\exists\mathbb{R}$-hard by giving an explicit polynomial-time reduction from bounded quartic equality feasibility. The reduction first transforms bounded quartic feasibility into homogeneous quadratic sphere feasibility by homogenization, box encoding, and quadratic lifting. It then maps the resulting homogeneous quadratic system to a quartic form whose maximum over the unit sphere separates feasible from infeasible instances. Finally, the quartic form is represented as a symmetric order-four tensor, yielding the desired tensor spectral threshold instance. The result shows that the computational obstruction in tensor spectral norm is not merely non-convex optimization or combinatorial hardness, but real algebraic feasibility itself.

翻译：我们从实代数复杂度的角度研究张量谱范数的判定版本。对于有理数指定的张量，张量谱阈值问题询问其谱范数是否超过一个预设的有理阈值。由于可行域是紧致的，极值的存在性本身是平凡的；有意义的是阈值判定问题。我们通过给出从有界四次等式可行性问题的显式多项式时间归约，证明该问题是$\exists\mathbb{R}$-难的。该归约首先通过齐次化、盒子编码和二次提升将有界四次可行性问题转化为齐次二次球面可行性问题，然后将所得的齐次二次系统映射为一个四次型，该四次型在单位球面上的最大值能够区分可行实例与不可行实例。最后，该四次型被表示为一个对称四阶张量，从而得到所需的张量谱阈值实例。该结果表明，张量谱范数中的计算障碍不仅仅是非凸优化或组合困难，而是实代数可行性本身。