Tensor slice rank and Cayley's first hyperdeterminant (Q2093514)

The authors study \(d\)-dimensional tensors \(T \in V^{\otimes d}\), i.e., multilinear elements of a vector space \(V \otimes \ldots \otimes V\). For a base \((e_i)\) each tensor has the coordinate representation \[ T = \sum_{i_1, \dots , i_d} T(i_1, \dots , i_d) \, \cdot \, e_{i_1} \otimes \ldots \otimes e_{i_d}. \] A \textit{simple tensor} is a non-zero tensor that is completely factorizable: \[ T (i_1, \dots , i_d) = v_1(i_1) \cdot \ldots \cdot v_d(i_d), \, \, \forall i_1, \dots i_d \in \{1 \dots n\}. \] Every tensor \(T\) can be expressed as a sum of simple tensors, and the minimum number of simple tensors that sum to \(T\) is known as the rank of \(T\). The main rank-properties of \(T\) are considered as follows: \begin{itemize} \item The \textit{slice rank} \(\operatorname{srank}(T)\) is defined if the simple tensors are decomposable along some coordinates: \[ T(i_1, \dots , i_d) = v(i_k) \, \cdot \, T(i_1, \dots , i_{k-1}, \, \, i_{k+1}, \dots, i_d), \forall i_1, \dots i_d \in \{1 \dots n\}. \] \item The \textit{partition rank} \(\operatorname{prank}(T)\) plays a role if simple tensors are decomposable with respect to a partitions of coordinates, i.e., \[ T(i_1, \dots , i_d) = T_1(i_{a_1}, \dots , i_{a_d}) \cdot T_2(i_{b_1}, \dots , i_{b_{d-k}}), \, \, \forall i_1, \dots i_d \in \{1 \dots n\} \] for some partitions \(\{a_1 < \dots < a_k\} \cup \{b_1 < \dots < b_{d-k}\}\) of the set \(\{ 1, \dots, d \}\). \item If one partition is of odd size, but does not contain the element \(1\), the authors define finally the \textit{odd partition rank} \(\operatorname{oprank}(T)\) of \(T\). \end{itemize} Along with these features the authors study the (first) hyperdeterminant of Cayley: \[ \det ( T ) = \sum_{\sigma_2, \dots, \sigma_d \in S_n} \operatorname{sgn}(\sigma_2 \dots \sigma_d) \cdot \prod_{i=1}^n T(i, \sigma_2(i), \dots, \sigma_d(i) ). \] The hyperdeterminant is linear, skew-symmetric and invariant under \(\operatorname{GL}(n)^d\). Using sum and product formulae for \(\det(T)\) the authors prove some theorems for the different versions of ranks, as for example: \begin{itemize} \item \(\operatorname{oprank}(T) < n \Rightarrow \det(T) = 0\), i.e., \(\det(T) \neq 0 \Rightarrow \operatorname{oprank}(T) = n\); \item \(\operatorname{rank}(T) \geq n\), and if \(d\) is even then \(\operatorname{srank}(T) = n\); \item A tensor over a field of characteristic \(p > 0\) with \(\operatorname{prank}(T) < \frac{n}{p-1}\) has \(\det(T) = 0\); thus \(\det(T) \neq 0 \Rightarrow \operatorname{prank}(T) \geq \frac{n}{p-1}\). \end{itemize} As an application the authors consider \(d\)-colored sum-ordered sets in \(\mathbb{F}_p^n\) with \(p\) prime and \(d > 2\). It is shown that its size \(N\) is \(N= \mathcal{O}(\gamma^n)\) for a constant \(1 \leq \gamma < p\).