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
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