16 the concept of a height [Chap. I
1.5. Remark. Let X
Pn
be a subvariety. Then every -rational point
on X is also a -rational point on Pn. We call the restriction of Hnaive to X( ) the
naive height on X. It is obvious that the naive height on X fulfills the fundamental
finiteness property.
1.6. Notation. i) For a prime number p, denote by .
p
the normalized p-
adic valuation. I.e., for x \{0}, let x =
±2v2 ·3v3
· . . . ·pkpk
v
be its decomposition
into prime factors. Then put
x
p
:=
p−vp
.
Further, 0
p
:= 0.
ii) We use .

as an alternative notation for the usual absolute value,
x

:=
x if x 0 ,
−x if x 0 .
1.7. Fact. For p a prime number or infinity, .
p
is indeed a valuation.
This means, for x, y ,
i) x
p
0,
ii) x
p
= 0 if and only if x = 0,
iii) xy
p
= x
p
· y
p
,
iv) x + y
p
x
p
+ y
p
.
For p = ∞, one even has that x + y
p
max{x
p
, y
p
}.
1.8. Fact (Product formula). For x \{0}, one has
p prime or
x
p
= 1 .
1.9. Lemma. Let (x0 : . . . : xn) Pn ( ). Then
Hnaive(x0 : . . . : xn) =
p prime or
max
i=0,...,n
xi
p
.
Proof. The product formula implies that the right-hand side remains unchanged
when (x0 : . . . : xn) is replaced by (λx0 : . . . : λxn) for λ = 0. Thus, we may
suppose that all xi are integers and gcd(x0, . . . , xn) = 1.
These assumptions imply that
max
i=0,...,n
xi
p
= 1
for every prime number p. Hence, the formula on the right-hand side may be
simplified to maxi=0,...,n |xi|. This is precisely the assertion.
1.10. Remark. Despite being so primitive, the naive height is actually suf-
ficient for most applications. For example, in the numerical experiments described
in Part C, we will always work with the naive height.
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