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 $2d$. Our methods may be of independent interest, and include a refined notion of polynomial decomposition that captures correlation with polynomials of lower degree than classical notions do, and a new correlation lemma that improves upon similar lemmas in the literature. Additionally, we illustrate the usefulness of the new correlation lemma by using it to give an alternative proof for the aforementioned result of Milićević and Randelović.

翻译：我们证明了在非小特征有限域上，对于次数为$d+1$的多项式，Gowers $U^d$范数的一个近多项式逆定理。关于次数$d$的情形，最近由Milićević和Randelović以完全多项式界解决。此外，我们给出了任意次数小于$2d$的齐次多项式的近多项式逆定理。我们的方法可能具有独立的研究价值，包括一种改进的多项式分解概念，该概念比经典概念更能捕捉与低次多项式的相关性，以及一个优于文献中类似引理的新相关性引理。进一步地，我们通过使用该新相关性引理为上述Milićević和Randelović的结果提供另一种证明，以此说明其实用性。