1.5. Inverse conjecture over finite fields 87
in the language of ergodic theory. However, in order not to have to introduce
too much additional material, we will describe the arguments here in the case
d = 3 without explicitly using ergodic theory notation. To do this, though,
we will have to sacrifice a lot of rigour and only work with some illustrative
special cases rather than the general case, and also use somewhat vague
terminology (e.g. “general position” or “low rank”).
To simplify things further, we will establish the U
inverse theorem
only for a special type of function, namely a quartic
e(φ), where
φ: V F is a classical polynomial of degree 4. The claim to show then
is that if e(φ)
U 3(V )
1, then e(φ) correlates with a cubic phase. In the
high characteristic case p 4, this result can be handled by equidistribution
theory. Indeed, since
U 3(V )
= Ex,h1,h2,h3,h4 e(∂h1 ∂h2 ∂h3 ∂h4 φ(x)),
that theory tells us that the quartic polynomial
(x, h1,h2,h3,h4) ∂h1 ∂h2 ∂h3 ∂h4 φ(x)
is low rank. On the other hand, in high characteristic one has the Taylor
φ(x) =
∂x∂x∂x∂xφ(0) + Q(x)
for some cubic function Q (as can be seen for instance by decomposing into
monomials). From this we easily conclude that φ itself has low rank (i.e.
it is a function of boundedly many cubic (or lower degree) polynomials), at
which point it is easy to see from Fourier analysis that e(φ) will correlate
with the exponential of a polynomial of degree at most 3.
Now we present a different argument that relies slightly less on the quar-
tic nature of φ; it is a substantially more difficult argument, and we will skip
some steps here to simplify the exposition, but the argument happens to ex-
tend to more general situations. As e(φ)
U 3
1, we have Δhe(φ)
U 2
for many h, thus by the inverse U
theorem, Δhe(φ) = e(∂hφ) correlates with
a quadratic phase. Using equidistribution theory, we conclude that the cubic
polynomial ∂hφ is low rank.
At present, the low rank property for ∂hφ is only true for many h. But
from the cocycle identity
(1.46) ∂h+kφ = ∂hφ + T
we see that if ∂hφ and ∂kφ are both low rank, then so is ∂h+kφ; thus the
property of ∂hφ being low rank is in some sense preserved by addition. Using
this and a bit of additive combinatorics, one can conclude that ∂hφ is low
14A good example to keep in mind is the symmetric polynomial phase e(S2/2) from Section
1.5.2, though one has to take some care with this example due to the low characteristic.
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