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Let $d \ge 3$ be an integer. We show that whenever an order-$d$ tensor admits$d+1$ decompositions according to Tao's slice rank, if the linear subspacesspanned by their one-variable functions constitute a sunflower for each choiceof special coordinate, then the tensor admits a decomposition where theselinear subspaces are contained in the centers of these respective sunflowers.As an application, we deduce that for every nonnegative integer $k$ and everyfinite field $\mathbb{F}$ there exists an integer $C(d,k,|\mathbb{F}|)$ suchthat every order-$d$ tensor with slice rank $k$ over $\mathbb{F}$ admits atmost $C(d,k,|\mathbb{F}|)$ decompositions with length $k$, up to a class oftransformations that can be easily described.

Small sunflowers and the structure of slice rank decompositions