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

  title={The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic},
  author={Terence Tao and Tamar D. Ziegler},
  journal={Annals of Combinatorics},
We establish the inverse conjecture for the Gowers norm over finite fields, which asserts (roughly speaking) that if a bounded function $${f : V \rightarrow \mathbb{C}}$$ on a finite-dimensional vector space V over a finite field $${\mathbb{F}}$$ has large Gowers uniformity norm $${{\parallel{f}\parallel_{U^{s+1}(V)}}}$$ , then there exists a (non-classical) polynomial $${P: V \rightarrow \mathbb{T}}$$ of degree at most s such that f correlates with the phase e(P) = e2πiP. This conjecture had… 
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