1.4. Equidistribution in finite fields 65
that P is constant on each coset of U. To put it another way, if we view U
as the intersection of a bounded number of kernels of linear homomorphism
L1,...,Ld : V → F (where d = O(1) is the codimension of U), then P
is constant on every simultaneous level set of L1,...,Ld, and can thus be
expressed as a function F (L1,...,Ld) of these linear polynomials.
More generally, let us say that a limit classical polynomial P of degree
≤ d is low rank if it can be expressed as P = F (Q1,...,Qd) where Q1,...,Qd
are a bounded number of polynomials of degree ≤ d − 1. We can summarise
the above discussion (and also Exercise 1.4.11) as follows:
Proposition 1.4.3. Let d ≤ 2, and let P : V → F be a limit classical
polynomial. If P is biased, then P is low rank.
In particular, from the Weyl criterion, we see that if P is not equidis-
tributed, then P is of low rank.
Of course, the claim fails if the low rank hypothesis is dropped. For
instance, consider a limit classical quadratic Q = L1L2 that is the product
of two linearly independent linear polynomials L1,L2. Then Q attains each
non-zero value with a density of (p −
rather than 1/p (and attains 0
with a density of (2p −
rather than 1/p).
Exercise 1.4.12. Suppose that the characteristic p of F is greater than
2, and suppose that P :
→ F is a quadratic polynomial of the form
P (x) =
x + c, where c ∈ F, b ∈
M is a symmetric n × n
matrix with coeﬃcients in F, and xT is the transpose of x. Show that
|Ex∈V e(P (x))| ≤
where r is the rank of M. Furthermore, if b is
orthogonal to the kernel of M, show that equality is attained, and otherwise
Ex∈V e(P (x)) vanishes.
What happens in the even characteristic case (assuming now that M is
Exercise 1.4.13 (Van der Corput lemma). Let P : V → F be a limit func-
tion on a limit finite dimensional vector space V , and suppose that there
exists a limit subset H of V which is sparse in the sense that |H| = o(|V |),
and such that ∂hP is equidistributed for all h ∈ V \H. Show that P itself is
equidistributed. Use this to give an alternate proof of Proposition 1.4.3.
Exercise 1.4.14 (Space of polynomials is discrete). Let P : V → F be a
polynomial of degree at most d such that Ex∈V |e(P (x)) − c|
some constant c ∈
Show that P is constant. (Hint: Induct on d.)
Conclude that if P, Q are two distinct polynomials of degree at most d, that
e(P ) − e(Q)
The fact that high rank polynomials are equidistributed extends to
higher degrees also: