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ACM2012-201-2P-T000 其他数据使用手册 - TDK（东电化）
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6 Terence Tao
Using the crude bound
d+n
n
≥
d
n
n
n
, we conclude as a coro llary that every finite
subset E of F
n
is contained in a F -hypersurface of degree at most n|E|
1/n
.
Proof. We re peat the second proof of Lemma 1.1(ii). If we let V be the vector
space of polynomials P ∈ F [x
1
, . . . , x
n
] of degree at most d, then a standard
combinatorial computation reveals that V has dimension
d+n
n
. If |E| <
d+n
n
,
then the linear map P 7→ (P (p))
p∈E
from V to F
E
thus has non-trivial kernel,
and the claim follows.
Example 1.5. If we set n = 2 a nd d equal to 1, 2 , or 3, then Lemma 1.4 makes
the fo llowing claims:
(1) Any two points in F
2
lie on a line;
(2) Any five points in F
2
lie on a (possibly degenerate) conic section; a nd
(3) Any nine points in F
2
lie on a (possibly degener ate) cubic curve.
Finally, we give a simple version (though certainly not the only version) of (c):
Lemma 1.6 (Dichotomy). Let F be a field, let n ≥ 1 be an integer, let Z(P ) be
a (geometric) hypersurface of degree at most d, and let ℓ be a (geometric) line.
Then either ℓ is (geometrically) contained in Z(P ), or else Z(P )[F ] ∩ ℓ[F ] has
cardinality at most d.
One can view this dichotomy as a rigidity statement: as soo n as a line meets
d + 1 or more po ints of a degree d hypersurface Z(P ), it must nec essarily “snap
into place” and beco me entirely contained (geometrically) inside that hypersurface .
These sorts of rigidity prope rties are a major source of powe r in the polynomial
method.
Proof. Write ℓ = {x
0
+ tv
0
: t ∈
F }, and then a pply Lemma 1.1(i) to the one-
dimensional polynomial t 7→ P (x
0
+ tv
0
).
As a quick application of these three lemmas, we give
Proposition 1.7 (Finite field Nikodym conjecture). Let F be a finite field, let
n, d ≥ 1 be integers and let E ⊂ F
n
have the property that through every point
x ∈ F
n
there exists a line ℓ
x,v
x
which contains more than d points from E. Then
|E| ≥
d+n
n
.
Proof. Clearly we may take d < |F |, as the hypothesis cannot be satisfied other-
wise. Suppose for contradiction that |E| <
d+n
n
; then by Lemma 1.4 one can
place E inside an F -hypersurface Z(P )[F ] of degree at most d. If x ∈ F
n
, then by
hypothesis there is a line ℓ
x,v
x
which meets E, and hence Z(P )[F ], in more than d
points; by Lemma 1.6, this implies that ℓ
x,v
x
is geometrically contained in Z(P ).
In particular, x lies in Z(P ) for every x ∈ F
n
, so in particular |Z(P )[F ]| = |F |
n
.
But this contradicts Le mma 1.2.
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