Small sunflowers and the structure of slice rank decompositions

Abstract We prove that for every integer , every nonnegative integer and every finite field there exists an integer such that every order‐ tensor with slice rank over admits at most decompositions with length , up to a class of transformations that can be easily described. A key result in the proof asserts that if an order‐ tensor admits slice rank decompositions and the linear subspaces spanned by their one‐variable functions constitute a sunflower for each choice of special coordinate, then the tensor admits a decomposition where these linear subspaces are contained in the centers of these respective sunflowers.