62 1. Higher order Fourier analysis
Exercise 1.4.3. Establish the
identity11
p(T
j
1) =
(−1)p−1(T j
1)(T 1)(T
2
1) . . . (T
p−1
1)
mod T
p
1
for an indeterminate T and any integer j, by testing on
pth
roots of unity.
Use this to give an alternate proof of Lemma 1.4.1.
This classifies all polynomials in the high characteristic case p d:
Corollary 1.4.2. If p d, then Poly≤d(V R/Z) = Poly≤d(V F) +
(R/Z). In other words, every polynomial of degree at most d is the sum of
a classical polynomial and a constant.
The situation is more complicated in the low characteristic case p d, in
which non-classical polynomials can occur (polynomials that are not simply
a classical polynomial up to constants). For instance, consider the function
P : F2 R/Z defined by P (0) = 0 and P (1) = 1/4. One easily verifies that
this is a (non-classical) quadratic (i.e., a polynomial of degree at most 2),
but is clearly not a shifted version of a classical polynomial since its range
is not a shift of the second roots {0, 1/2} mod 1 of unity.
Exercise 1.4.4. Let P : F2
R/Z be a function. Show that P is a poly-
nomial of degree at most d if and only if the range of P is a translate of the
(2d)th
roots of unity (i.e.,
2dP
is constant).
For further discussion of non-classical polynomials, see [Ta2009, §1.12].
Henceforth we shall avoid this technical issue by restricting to the high
characteristic case p d (or equivalently, the low degree case d p).
1.4.2. Equidistribution. Let us now consider the equidistribution theory
of a classical polynomial P : V F, where we think of F as being a fixed
field (in particular, p = O(1)), and the dimension of V as being very large; V
will play the role here that the interval [N] played in Section 1.1. This the-
ory is classical for linear and quadratic polynomials. The general theory was
studied first in [GrTa2009] in the high characteristic case p d, and ex-
tended to the low characteristic case in [KaLo2008]; see also [HaSh2010],
[HaLo2010] for some recent refinements. An analogous theory surely exists
for the non-classical case, although this is not currently in the literature.
The situation here is simpler because a classical polynomial can only
take p values, so that in the equidistributed case one expects each value
to be obtained about |V |/p times. Inspired by this, let us call a classical
polynomial P δ-equidistributed if one has
|{x V : P (x) = a} |V |/p| δ|V |
for all a F.
11We
thank Andrew Granville for showings us this argument.
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