62 1. Higher order Fourier analysis

Exercise 1.4.3. Establish the

identity11

p(T

j

− 1) =

(−1)p−1(T j

− 1)(T − 1)(T

2

− 1) . . . (T

p−1

− 1)

mod T

p

− 1

for an indeterminate T and any integer j, by testing on

pth

roots of unity.

Use this to give an alternate proof of Lemma 1.4.1.

This classifies all polynomials in the high characteristic case p d:

Corollary 1.4.2. If p d, then Poly≤d(V → R/Z) = Poly≤d(V → F) +

(R/Z). In other words, every polynomial of degree at most d is the sum of

a classical polynomial and a constant.

The situation is more complicated in the low characteristic case p ≤ d, in

which non-classical polynomials can occur (polynomials that are not simply

a classical polynomial up to constants). For instance, consider the function

P : F2 → R/Z defined by P (0) = 0 and P (1) = 1/4. One easily verifies that

this is a (non-classical) quadratic (i.e., a polynomial of degree at most 2),

but is clearly not a shifted version of a classical polynomial since its range

is not a shift of the second roots {0, 1/2} mod 1 of unity.

Exercise 1.4.4. Let P : F2

→ R/Z be a function. Show that P is a poly-

nomial of degree at most d if and only if the range of P is a translate of the

(2d)th

roots of unity (i.e.,

2dP

is constant).

For further discussion of non-classical polynomials, see [Ta2009, §1.12].

Henceforth we shall avoid this technical issue by restricting to the high

characteristic case p d (or equivalently, the low degree case d p).

1.4.2. Equidistribution. Let us now consider the equidistribution theory

of a classical polynomial P : V → F, where we think of F as being a fixed

field (in particular, p = O(1)), and the dimension of V as being very large; V

will play the role here that the interval [N] played in Section 1.1. This the-

ory is classical for linear and quadratic polynomials. The general theory was

studied first in [GrTa2009] in the high characteristic case p d, and ex-

tended to the low characteristic case in [KaLo2008]; see also [HaSh2010],

[HaLo2010] for some recent refinements. An analogous theory surely exists

for the non-classical case, although this is not currently in the literature.

The situation here is simpler because a classical polynomial can only

take p values, so that in the equidistributed case one expects each value

to be obtained about |V |/p times. Inspired by this, let us call a classical

polynomial P δ-equidistributed if one has

|{x ∈ V : P (x) = a} − |V |/p| ≤ δ|V |

for all a ∈ F.

11We

thank Andrew Granville for showings us this argument.