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The regularity lemma is false over small fields

Abstract The regularity lemma is a stringent condition of the possible ranks of tensor blow-ups of linear subspaces of matrices. It was proved by Ivanyos, Qiao and Subrahmanyam in [5] when the underlying field is sufficiently large. We show that if the field size is too small, the regularity lemma is false – this settles an open problem posed by Ivanyos, Qiao and Subrahmanyam.