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We establish the \emph{inverse conjecture for the Gowers norm over finitefields}, which asserts (roughly speaking) that if a bounded function $f: V \to\C$ on a finite-dimensional vector space $V$ over a finite field $\F$ has largeGowers uniformity norm $\|f\|_{U^{s+1}(V)}$, then there exists a(non-classical) polynomial $P: V \to \T$ of degree at most $s$ such that $f$correlates with the phase $e(P) = e^{2\pi i P}$. This conjecture had alreadybeen established in the ``high characteristic case'', when the characteristicof $\F$ is at least as large as $s$. Our proof relies on the weak form of theinverse conjecture established earlier by the authors and Bergelson, togetherwith new results on the structure and equidistribution of non-classicalpolynomials, in the spirit of the work of Green and the first author and ofKaufman and Lovett.

The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic