## Ranks of tensors and polynomials, with combinatorial applications

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## Abstract

This thesis will consist of three main chapters.

It is a standard fact of linear algebra that every matrix with rank k contains a k ⨯ k submatrix with rank k. In Chapter 2 we generalise this fact asymptotically to a class of notions of rank for higher-order tensors, containing in particular the tensor rank, the slice rank and the partition rank. We show that for every integer d ⩾ 2 and every notion R in this class of notions of rank, there exist functions F

In Chapter 3 we extend to the case of restricted subsets a result of Green and Tao on the equidistribution of high-rank polynomials over finite prime fields. We show that for every fixed prime integer p, for every integer d ∈ [2, p-1], and for every non-empty subset S of F

In Chapter 4 we prove approximation results for conditions on {0,1}