## The distribution of polynomials over finite fields, with applications to the Gowers norms

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Green, Ben

Tao, Terence

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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 establish an
inverse result for the Gowers U^{d+1}-norm of functions of the form f(x)=
e_F(P(x)), where P : F^n -> F is a polynomial of degree less than F, showing
that this norm can only be large if f correlates with e_F(Q(x)) for some
polynomial Q : F^n -> F of degree at most d.
The requirement deg(P) < |F| cannot be dropped entirely. Indeed, we show the
above claim fails in characteristic 2 when d = 3 and deg(P)=4, showing that the
quartic symmetric polynomial S_4 in F_2^n has large Gowers U^4-norm but does
not correlate strongly with any cubic polynomial. This shows that the theory of
Gowers norms in low characteristic is not as simple as previously supposed.
This counterexample has also been discovered independently by Lovett, Meshulam,
and Samorodnitsky.
We conclude with sundry other applications of our main result, including a
recurrence result and a certain type of nullstellensatz.

Comment: 33 pages

Comment: 33 pages

##### Keywords

Mathematics - Combinatorics, Mathematics - Number Theory