1.4. Equidistribution in finite fields 73

finite (or limit finite) for d 2. (For d = 1, the analytic rank is infinite if P

is non-constant and zero if P is constant.)

Exercise 1.4.19. Show that if p 2 and P is a (limit) classical polynomial

of degree 2, then rank1(P ) = arank1(P ).

Exercise 1.4.20. Show that if the analytic rank arankd−1(P ) of a limit

classical polynomial P of degree d is unbounded, then P is equidistributed.

Exercise 1.4.21. Suppose we are in the high characteristic case p d.

Using Proposition 1.4.3, show that a limit classical polynomial has bounded

analytic rank if and only if it has bounded rank. (Hint: One direction

follows from the preceding exercise. For the other direction, use the Taylor

formula P (x) =

1

d!

∂xP

d

(x).) This is a special case of the inverse conjecture

for the Gowers norms, which we will discuss in more detail in later sections.

Conclude the following finitary version: if P : V → F is a classical poly-

nomial of degree d on a finite-dimensoinal vector space V , and arankd−1(P )

≤ M, then rankd−1(P )

M,p,d

1; conversely, if rankd−1(P ) ≤ M, then

arankd−1(P )

M,p,d

1.

Exercise 1.4.22. Show that if P is a (limit) classical polynomial of degree

d, then rankd−1(P ) = rankd−1(cP ) and arankd−1(P ) = arankd−1(cP ) for

all c ∈ F\0, and rankd−1(P + Q) = rankd−1(P ) and arankd−1(P + Q) =

arankd−1(P ) for all (limit) classical polynomials Q of degree ≤ d − 1.

It is clear that the rank obeys the triangle inequality rankd−1(P + Q) ≤

rankd−1(P ) + rankd−1(Q) for all (limit) classical polynomials of degree ≤ d.

There is an analogue for analytic rank:

Proposition 1.4.11 (Quasi-triangle inequality for analytic rank). (See

[GoWo2010b].) Let P, Q: V → F be (limit) classical polynomials of degree

≤ d. Then arankd−1(P + Q) ≤

2d(arankd−1(P

) + arankd−1(Q)).

Proof. Let T1(h1,...,hd) be the d-linear form

T1(h1,...,hd) := ∂h1 . . . ∂hd P (x)

(note that the right-hand side is independent of x); similarly define

T2(h1,...,hd) := ∂h1 . . . ∂hd P (x).

By definition, we have

Eh1,...,hd∈V e(T1(h1,...,hd)) =

p− arankd−1(P )

and

Eh1,...,hd∈V e(T2(h1,...,hd)) =

p− arankd−1(Q)