Stronger arithmetic equivalence

Shachar Lovett

doi:10.19086/da.8654

Abstract

This paper considers two alternative strengthenings of the notion of arithmetic equivalence, which the author calls local integral equivalence and solvable equivalence. (The latter turns out to be strictly stronger than the former.) They have the advantage of being easier to check than Prasad’s notion, which the author calls integral equivalence. Furthermore, solvable equivalence, which the author shows does not imply integral equivalence, is nevertheless a sufficient condition to imply that the invariants considered by Prasad are equal. This opens the door to easier proofs of Prasad’s result, and lessens the reliance on Scott’s construction: the author finds a generalization of this construction that yields infinitely many examples of solvable equivalence. The paper also contains several examples to clarify the relationships between the various different notions of equivalence. Some of these examples (which are mainly found with the help of a computer) answer open questions from the group theory literature.

Highlights

The main objects of study in this paper are tensors, or equivalently multilinear forms
More relevant to the topic of the current paper, recently defined notions of ranks were instrumental in the resolution of the cap-set problem in additive combinatorics [9, 10, 25], which is in itself intimately related to the problem of matrix multiplication [1, 7]
One of the results of this paper is that the analytic rank can replace the role played by the slice rank or partition rank in the current proofs, and it is always a lower bound for these latter ranks

Summary

Introduction

The main objects of study in this paper are tensors, or equivalently multilinear forms. The analytic rank lower bounds all the previously known combinatorial notions of rank (standard one, slice rank and partition rank). One of the results of this paper is that the analytic rank can replace the role played by the slice rank or partition rank in the current proofs, and it is always a lower bound for these latter ranks. This raises an alternative approach to using tensor-rank based techniques to study these Ramsey problems

Tensors and tensor ranks

The analytic rank

Applications

Is the analytic rank really better than the partition rank?