Nearly-polynomial inverse theorem for the U^d norm in degree d+1

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 Mili\'{c}evi\'{c} and Randelovi\'{c} 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 Mili\'{c}evi\'{c} and Randelovi\'{c}.