1.4. Equidistribution in finite fields 63

Exercise 1.4.5. Show that this is equivalent to the notion of δ-equidistri-

bution given in Section 1.1, if one gives F the metric induced from R/Z,

and if one is willing to modify δ by a multiplicative factor depending on p

in the equivalences.

Before we study equidistribution in earnest, we first give a classical es-

timate.

Exercise 1.4.6 (Chevalley-Warning theorem). Let V be a finite dimensional

space, and let P : V → F be a classical polynomial of degree less than

(p − 1) dim(V ). Show that

∑

x∈V

P (x) = 0. (Hint: Identify V with

Fn

for some n and apply Exercise 1.4.1. Use the fact that

∑

x∈F

xi

= 0 for

all 1 ≤ i p − 1, which can be deduced by using a change of variables

x → bx.) If, furthermore, P has degree less than dim(V ), conclude that for

every a ∈ F, that |{x ∈ V : P (x) = a}| is a multiple of p. (Hint: Apply

Fermat’s little theorem to the quantity (P

−a)p−1.)

In particular, if x0 ∈ V ,

then there exists at least one further x ∈ V such that P (x) = P (x0).

If P has degree at most d and x0 ∈ V , obtain the recurrence inequality

|{x ∈ V : P (x) = P (x0)}|

p,d

|V |.

(Hint: Normalise x0 = 0, then average the previous claim over all subspaces

of V of a certain dimension.)

The above exercise goes some way towards establishing equidistribution,

by showing that every element in the image of P is attained a fairly large

number of times. But additional techniques will be needed (together with

additional hypotheses on P ) in order to obtain full equidistribution. It will

be convenient to work in the ultralimit setting. Define a limit classical poly-

nomial P : V → F on a limit finite-dimensional vector space V =

α→α∞

Vα

of degree at most d to be an ultralimit of classical polynomials Pα : Vα → F

of degree at most d (we keep F and d fixed independently of α). We say

that a limit classical polynomial P is equidistributed if one has

|{x ∈ V : P (x) = a}| = |V |/p + o(|V |)

for all a ∈ F, where the cardinalities here are of course limit cardinalities.

Exercise 1.4.7. Let V be a limit finite-dimensional vector space. Show that

a limit function P : V → F is a limit classical polynomial of degree at most

d if and only if it is a classical polynomial of degree at most d (observing

here that every limit vector space is automatically a vector space).

Exercise 1.4.8. Let P = limα→α∞ Pα be a limit classical polynomial.

Show that P is equidistributed if and only if, for every δ 0, Pα is δ-

equidistributed for α suﬃciently close to α∞.