Theory of Computing ------------------- Title : Inverse Conjecture for the Gowers Norm is False Authors : Shachar Lovett, Roy Meshulam, and Alex Samorodnitsky Volume : 7 Number : 9 Pages : 131-145 URL : https://theoryofcomputing.org/articles/v007a009 Abstract -------- Let p be a fixed prime number and N be a large integer. The "Inverse Conjecture for the Gowers norm" states that if the "d-th Gowers norm" of a function f : F_p^N \to F_p is non-negligible, that is, larger than a constant independent of N, then f is non-trivially correlated to 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 d = 4, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation to any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao (2009).