A Generalization of the Subspace Theorem With Polynomials of Higher Degree

Abstract

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 ∈ℙⁿ(K) of

$$ \log \left( {\prod\limits_{v \in S} {\prod\limits_{i = 0}^n {\frac{{\left| {L_i^{\left( v \right)} \left( x \right)} \right|_v }} {{\left\| x \right\|_v }}} } } \right) \leqslant - \left( {n + 1 + \delta } \right)h\left( x \right) $$

((1.1))

is contained in the union of finitely many proper linear subspaces of ℙⁿ.