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THE DISTRIBUTION OF POLYNOMIALS OVER FINITE FIELDS, WITH APPLICATIONS TO THE GOWERS NORMS
In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P : F n → F is poorly-distributed only if P is determined by the values of a few polynomials of lower degree, in which case we say that P has small rank. We give several applications of this result, paying particular attention to consequences for the theory of the so-called Gowers norms. We establishfatcat:zyygvmofabfmrasrkpcqywblkq