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Published in:
2008 | OriginalPaper | Chapter
A Generalization of the Subspace Theorem With Polynomials of Higher Degree
Authors : Jan-Hendrik Evertse, Roberto G. Ferretti
Published in: Diophantine Approximation
Publisher: Springer Vienna
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1.1 The Subspace Theorem can be stated as follows. Let
K
be a number field (assumed to be contained in some given algebraic closure
of ℚ),
n
a positive integer, 0 < δ
≤
1 and
S
a finite set of places of
K.
For
v ∈
S
, let
$$ L_0^{\left( v \right)} , \ldots ,L_n^{\left( v \right)} $$
be linearly independent linear forms in
[
x
0
,...,
x
n
]. Then the set of solutions
x
∈ℙ
n
(
K
) of
(1.1)
$$ \log \left( {\prod\limits_{v \in S} {\prod\limits_{i = 0}^n {\frac{{\left| {L_i^{\left( v \right)} \left( x \right)} \right|_v }} {{\left\| x \right\|_v }}} } } \right) \leqslant - \left( {n + 1 + \delta } \right)h\left( x \right) $$
is contained in the union of finitely many proper linear subspaces of ℙ
n
.