Authors: Milićević, Luka 
Affiliations: Mathematics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Approximately symmetric forms far from being exactly symmetric
Journal: Combinatorics, Probability and Computing
Volume: 32
Issue: 2
First page: 299
Last page: 315
Issue Date: 2023
Rank: ~M22
DOI: 10.1017/S0963548322000244
Let V be a finite-dimensional vector space over Fp . We say that a multilinear form α:Vk→Fp in k variables is d -approximately symmetric if the partition rank of difference α(x1,…,xk)−α(xπ(1),…,xπ(k)) is at most d for every permutation π∈Symk . In a work concerning the inverse theorem for the Gowers uniformity ‖⋅‖U4 norm in the case of low characteristic, Tidor conjectured that any d -approximately symmetric multilinear form α:Vk→Fp differs from a symmetric multilinear form by a multilinear form of partition rank at most Op,k,d(1) and proved this conjecture in the case of trilinear forms. In this paper, somewhat surprisingly, we show that this conjecture is false. In fact, we show that approximately symmetric forms can be quite far from the symmetric ones, by constructing a multilinear form α:Fn2×Fn2×Fn2×Fn2→F2 which is 3-approximately symmetric, while the difference between α and any symmetric multilinear form is of partition rank at least Ω(3√n) .
Keywords: Multilinear forms | Partition rank | Symmetric group
Publisher: Cambridge University Press
Project: This work was supported by the Serbian Ministry of Education, Science and Technological Development through Mathematical Institute of the Serbian Academy of Sciences and Arts.

Files in This Item:
File Description SizeFormat
LMilicevic.pdf372.26 kBAdobe PDFView/Open
Show full item record

Page view(s)

checked on May 9, 2024


checked on May 9, 2024

Google ScholarTM




This item is licensed under a Creative Commons License Creative Commons