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We prove quantitative bounds for the inverse theorem for Gowers uniformitynorms $\mathsf{U}^5$ and $\mathsf{U}^6$ in $\mathbb{F}_2^n$. The proof startsfrom an earlier partial result of Gowers and the author which reduces theinverse problem to a study of algebraic properties of certain multilinearforms. The bulk of the work in this paper is a study of the relationshipbetween the natural actions of $\operatorname{Sym}_4$ and$\operatorname{Sym}_5$ on the space of multilinear forms and the partitionrank, using an algebraic version of regularity method. Along the way, we give apositive answer to a conjecture of Tidor about approximately symmetricmultilinear forms in 5 variables, which is known to be false in the case of 4variables. Finally, we discuss the possible generalization of the argument for$\mathsf{U}^k$ norms.

Quantitative inverse theorem for Gowers uniformity norms $\mathsf{U}^5$ and $\mathsf{U}^6$ in $\mathbb{F}_2^n$