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∞

of degree at most d to be an ultralimit of classical polynomials : 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∞ be a limit classical polynomial.
Show that P is equidistributed if and only if, for every δ 0, is δ-
equidistributed for α sufficiently close to α∞.
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