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 −

1)/p2

rather than 1/p (and attains 0

with a density of (2p −

1)/p2

rather than 1/p).

Exercise 1.4.12. Suppose that the characteristic p of F is greater than

2, and suppose that P :

Fn

→ F is a quadratic polynomial of the form

P (x) =

xT

Mx +

bT

x + c, where c ∈ F, b ∈

Fn,

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))| ≤

p−r/2,

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

not symmetric)?

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|

2−d+1

for

some constant c ∈

S1.

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)

L2(V )

1.

The fact that high rank polynomials are equidistributed extends to

higher degrees also: