We prove a nearly polynomial inverse theorem for the Gowers $U^d$ norm, over finite fields of non-small characteristic, for polynomials of degree $d+1$. The case of degree $d$ was very recently settled by Milićević and Randelović with a fully polynomial bound. We moreover provide a nearly polynomial inverse theorem for homogeneous polynomials of any degree smaller than $2d$.

翻译：我们证明了在非小特征有限域上，对于 $d+1$ 次多项式，Gowers $U^d$ 范数的一个近多项式逆定理。关于 $d$ 次多项式的相应结果近期由 Milićević 和 Randelović 解决，他们给出了一个完全多项式界。此外，我们为任意次数小于 $2d$ 的齐次多项式提供了一个近多项式逆定理。