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ACM2012-201-2P-T000 其他数据使用手册 - TDK（东电化）
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ACM2012-201-2P-T000 200Ω/OHM 0805-201
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Polynomial method in combinatorics 3
formal expressio n P (x
1
, . . . , x
n
) of the form
P (x
1
, . . . , x
n
) =
X
i
1
,...,i
n
≥0
c
i
1
,...,i
n
x
i
1
1
. . . x
i
n
n
where the coefficients c
i
1
,...,i
n
lie in F , and only finitely many of the coefficients
are non-zero. The degree of this polynomial is the largest value of i
1
+ . . . + i
n
for which c
i
1
,...,i
n
is non-zero; we will adopt the convention that the z ero polyno-
mial (which we will also call the trivial polynomial) has degree −∞. The space
of F -polynomials in n variables will be denoted F [x
1
, . . . , x
n
]. This space is of
course contained in the larger space
F [x
1
, . . . , x
n
] of geometric polynomials whose
coefficients now lie in F , but we will rarely need to use this space.
Of c ourse, by interpreting the indeterminate variables x
1
, . . . , x
n
as elements
of F , we can view an F -po lynomial P ∈ F [x
1
, . . . , x
n
] as a function from F
n
to
F ; it may also be viewed as a function from
F
n
to F . By abuse
3
of notatio n, we
denote both of these functions P : F
n
→ F and P :
F
n
→ F by P . This defines
two closely related sets, the geometric hypersurface
Z(P ) = Z(P )[
F ] := {(x
1
, . . . , x
n
) ∈ F
n
: P (x
1
, . . . , x
n
) = 0 }
and the F - hypersur face
Z(P )[F ] := {(x
1
, . . . , x
n
) ∈ F
n
: P (x
1
, . . . , x
n
) = 0 }
(also known as the set of F-p oints of the geometric hyp ersurface). We say that
the geometric hypersurface Z(P ) has degree d if P has degree d. More generally,
given a collection P
1
, . . . , P
k
∈ F [x
1
, . . . , x
n
] of polynomials, we may form the
4
geometric variety
Z(P
1
, . . . , P
k
) = Z(P
1
, . . . , P
k
)[
F ] =
k
\
i=1
Z(P
i
)[
F ]
and the F - variety
Z(P
1
, . . . , P
k
)[F ] =
k
\
i=1
Z(P
i
)[F ]
cut out by the k polynomials P
1
, . . . , P
k
. For instance, if x
0
, v
0
∈ F
n
with v
0
non-zero, the geometric line
ℓ
x
0
,v
0
= ℓ
x
0
,v
0
[
F ] := {x
0
+ tv
0
: t ∈ F }
3
One should caution though that two polynomials may be different even if they define the s ame
function from F
n
to F . For instance, i f F = F
q
is a finite field, the polynomials x
q
and x in
F [x] give rise to the same function from F to F , but are not the same polynomial (note for
instance that they have different degree). On the other hand, this ambiguity does not occur
in the algebraic closure
F , whi ch is necessarily infinite; thus, if one w ishes, one may identify
P with the function P :
F
n
→ F , but not necessarily with the function P : F
n
→ F (unless
F is infinite or P has degree less than |F |, in which case no ambiguity occurs, thanks to the
Schwartz-Zippel lemma (see Lemma 1.2 below)).
4
In this survey we do not requir e varieties to be irreducible.
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