A quasipolynomial inverse theorem for the $\mathsf{U}^k(\mathbb{F}_p^n)$ norm in the high characteristic

We prove an inverse theorem for the Gowers uniformity norm $\mathsf{U}^k(\mathbb{F}_p^n)$ with quasipolynomial bounds in the case when $p \geq k$. The inverse theorem follows from a quasipolynomial structure theorem for Freiman multihomomorphisms, which are a natural generalization of Freiman homomo