On the Slice Rank of Tensors in P-Echelon Form

Technical Summary: On the Slice Rank of Tensors in P-Echelon Form

Problem Statement
The paper addresses the problem of determining the slice-rank of $d$-tensors $T: A^d \to \mathbb{F}$, where $A$ is a totally ordered finite set and $\mathbb{F}$ is a field. Specifically, it investigates tensors in $P$-echelon form, defined relative to a partially ordered set (poset) $P = ([d], \leq_P)$. A tensor is in $P$-echelon form if, for every tuple $(a_1, \dots, a_d)$ in its support, the condition $r \leq_P s$ implies $a_r \leq_Q a_s$, where $\leq_Q$ is the total order on $A$.

The central question is whether such tensors, provided they have nonzero diagonal entries (i.e., $T(a, \dots, a) \neq 0$ for all $a \in A$), possess full slice-rank, meaning $\text{sr}(T) = |A|$. This generalizes Tao's slice-rank lemma for diagonal tensors. Previous work by Amanov and Yeliussizov established this result for even $d$ under the stricter condition that the Hasse diagram of $P$ is connected.

Methodology
The authors employ a functional reformulation of a proposition by Sawin and Tao to establish a lower bound on the slice-rank. The core of the proof relies on Lemma 2.1 (Sawin-Tao), which states that the slice-rank is bounded below by the minimum sum of the sizes of projections of a partition of the set of maximal elements of the tensor's support.

To apply this lemma, the authors introduce a specific ordering strategy in Lemma 2.4. They demonstrate that for any poset $P$ whose Hasse diagram contains no isolated vertices, one can assign either the original order $\leq_Q$ or the reverse order $\geq_Q$ to each coordinate $j \in [d]$. This assignment ensures that if a diagonal element $\delta_a = (a, \dots, a)$ is less than or equal to a support element $x$ under the resulting product order, then $x$ must equal $\delta_a$. This property forces the set of diagonal elements to be contained within the set of maximal elements of the support.

Key Contributions and Results
The paper presents Theorem 1.3, the main result, which asserts:

Let $d \geq 2$. Let $A$ be a totally ordered finite set, $\mathbb{F}$ a field, and $P = ([d], \leq_P)$ a poset whose Hasse diagram has no isolated vertex. If $T: A^d \to \mathbb{F}$ is in $P$-echelon form with nonzero diagonal entries, then $\text{sr}(T) = |A|$.

The proof proceeds by:

1. Establishing that the diagonal elements $\text{Diag}(A^d)$ are a subset of the maximal elements $\Gamma$ of the support under the constructed product order.
2. Showing that for any partition of $\Gamma$ into $d$ sets, the sum of the sizes of their coordinate projections is at least $|A|$.
3. Combining this lower bound with the trivial upper bound (achieved by slicing along the first coordinate) to conclude equality.

Significance and Claims
The authors explicitly frame their contribution as an extension and improvement of the results by Amanov and Yeliussizov. The significance is twofold:

1. Parity Independence: The result holds for both even and odd $d$, whereas the previous result by Amanov and Yeliussizov was restricted to even $d$.
2. Relaxed Connectivity Assumption: The paper replaces the requirement that the Hasse diagram of $P$ be connected with the weaker condition that it has no isolated vertex.

The paper maintains a modest scope, focusing strictly on the theoretical determination of slice-rank for this specific class of tensors. It does not propose new applications, experimental validations, or future implications beyond the mathematical generalization of the existing lemma within the context of the slice-rank method in extremal combinatorics.