72 1. Higher order Fourier analysis

R1,...,Rm generates a finite-dimensional space here, which thus has a ba-

sis Sd−1,1,...,Sd−1,md−1 mod Polyd−1,

0

thus Sd−1,1,...,Sd−1,md−1 are lin-

early independent modulo low rank polynomials of degree d − 1, and the

R1,...,Rm are linear combinations of the Sd−1,1,...,Sd−1,md−1 plus combi-

nations of some additional polynomials R1,...,Rm of degree d−2. Applying

the induction hypothesis to those additional polynomials, one obtains the

claim.

Exercise 1.4.18. Show that the polynomials S := (Sd ,i)1≤d

≤d−1;1≤i≤m

d

appearing in the above lemma are equidistributed in the sense that

|{x ∈ V : S(x) = s}| =

1

p

∑d

d =1

md

+ o(1) |V |

for any s = (sd ,i)1≤d

≤d−1;1≤i≤md

with sd

,i

∈ F.

Applying the above lemma, one can express any order d function Q

in the form Q = F ((Sd ,i)1≤d

≤d−1;1≤i≤md

). It is then possible to modify

the previous arguments to obtain Proposition 1.4.6; see [GrTa2009] for

more details. (We phrase the arguments in a finitary setting rather than a

non-standard one, but the two approaches are equivalent; see Section 2.1 for

more discussion.)

It is possible to modify the above arguments to handle the low char-

acteristic case, but due to the lack of a good Taylor expansion, one has

to regularise the derivatives of the polynomials, as well as the polynomials

themselves; see [KaLo2008] for details.

1.4.3. Analytic rank. Define the rank rankd−1(P ) of a degree d (limit)

classical polynomial P to be the least number m of degree ≤ d − 1 (limit)

classical polynomials R1,...,Rm such that P is a function of R1,...,Rm.

Proposition 1.4.3 tells us that P is equidistributed whenever the rank is

unbounded. However, the proof was rather involved. There is a more ele-

mentary approach to equidistribution to Gowers and Wolf [GoWo2010b]

which replaces the rank by a different object, called analytic rank, and which

can serve as a simpler substitute for the concept of rank in some applications.

Definition 1.4.10 (Analytic rank). The analytic rank arankd−1(P ) of a

(limit) classical polynomial P : V → F of degree ≤ d is defined to be the

quantity

arankd(P ) := − logp Ex,h1,...,hd∈V e(∂h1 . . . ∂hd P (x))

=

−2d

logp e(P )

Ud(V )

.

From the properties of the Gowers norms we see that this quantity is

non-negative, is zero if and only if P is a polynomial of degree d, and is