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Let $p$ be a fixed prime number, and $N$ be a large integer. The 'InverseConjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of afunction $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constantindependent of $N$, then $f$ can be non-trivially approximated by a degree$d-1$ polynomial. The conjecture is known to hold for $d=2,3$ and for any prime$p$. In this paper we show the conjecture to be false for $p=2$ and for $d =4$, by presenting an explicit function whose 4-th Gowers norm isnon-negligible, but whose correlation any polynomial of degree 3 isexponentially small.  Essentially the same result (with different correlation bounds) wasindependently obtained by Green and Tao \cite{gt07}. Their analysis uses amodification of a Ramsey-type argument of Alon and Beigel \cite{ab} to showinapproximability of certain functions by low-degree polynomials. We observethat a combination of our results with the argument of Alon and Beigel impliesthe inverse conjecture to be false for any prime $p$, for $d = p^2$.

Inverse Conjecture for the Gowers norm is false