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.

Keywords:

• Regularity lemma
• tensor blow-ups
• Higman's trick

2010 Mathematical Subject Classification:

• 16R50
• 17A35
• 15A69

Additional information

Funding

The first author was supported by NSF Grant No. DMS-1601229. The second author was supported by NSF Grant No. DMS-1601229, DMS-1638352 and CCF-1412958.