We show that determining the rank of a tensor over a field has the same complexity as deciding the existential theory of that field. This implies earlier NP-hardness results by H{\aa}stad~\cite{H90}. The hardness proof also implies an algebraic universality result.

翻译：我们证明，在域上确定张量的秩与该域的存在性理论判定具有相同的复杂度。这一结果蕴含了H{\aa}stad~\cite{H90}先前的NP困难性结论。该困难性证明还蕴含了一个代数普适性结果。