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ACM2012-201-2P-T000 其他数据使用手册 - TDK（东电化）
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4 Terence Tao
is a geometric variety (cut out by n − 1 affine-linear polynomials), and similarly
the F -line
ℓ
x
0
,v
0
[F ] = {x
0
+ tv
0
: t ∈ F }
is the associated F -variety.
When the ambient dimension n is equal to 1, F -hypersurfac es can be described
exactly:
Lemma 1.1 (Hypersurfaces in one dimension). Let d ≥ 0.
(i) (Factor theorem) If P ∈ F [x] is a non-trivial polynomial of degree at most
d, then Z(P )[F ] has cardinality at most d.
(ii) (Interpolation theorem) Conversely, if E ⊂ F has cardinality at most d, then
there is a non-trivial polynomial P ∈ F [x] with E ⊂ Z(P )[F ].
Proof. If Z(P )[F ] contains a point p, then P factors as P (x) = (x − p)Q(x) for
some polynomial Q of degree at most d−1, and (i) follows from induction on d. For
(ii), one c an simply take P (x) :=
Q
p∈E
(x − p). Alternatively, one can use linear
algebra: the space of polyno mials P of degree at most d is a d + 1-dimensional
vector space over F , while the space F
E
of tuples (y
p
)
p∈E
is at most d dimensional.
Thus, the evaluation map P 7→ (P (p))
p∈E
between these two spaces must have a
non-trivial kernel, and (ii) follows.
While these one-dimensional facts are almost trivial, they do illustr ate three
basic phenomena:
(a) “Low-degree” F -hypersurfaces tend to be “small” in a combinatorial sense.
(b) Conversely, “small” combinatorial sets tend to be captured by “low-degree”
F -hype rsurfaces.
(c) “Low-complexity” F -algebraic sets (such as {x ∈ F : P (x) = 0}) tend
to exhibit size dichotomies; either they are very small or very large (e.g.
{x ∈ F : P (x) = 0} is very small when P is non-zero and very large when P
is zero).
These phenomena become much more interesting and powerful in higher di-
mensions. Fo r instanc e, we have the following higher-dimensional version of (a):
Lemma 1.2 (Schwartz-Zippel lemma ). [59, 87] Let F be a finite field, let n ≥ 1,
and let P ∈ F [x
1
, . . . , x
n
] be a polynomial of degree at most d. If P does not vanish
entirely, then
|Z(P )[F ]| ≤ d|F |
n−1
.
Proof. This will be an iterated version of the argument used to prove Lemma
1.1(i). We induct on the dimension n. The case n = 1 follows from Lemma 1.1(i),
so suppose inductively that n > 1 a nd that the claim has alre ady been proven for
n − 1.
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