Roberto G. Ferretti

Diophantine inequalities on projective varieties

then the set of solutions of (1.1) lies in the union of finitely many proper linear subspaces of P. We give an equivalent formulation on which we shall focus in this paper. Let {l0, . . . , lN} be the union of the sets {l0v, . . . , lnv} (v ∈ S). Define the map φ : P → P by y 7→ ( l0(y) : · · · : lN(y) ) . Put X := φ(P); then X is a linear subvariety of P of dimension n defined over K. Write xi = li(y) (i = 0, . . . , N), x = (x0 : · · · : xN) = φ(y). For v ∈ S, let Iv be the set of indices given by {li : i ∈ Iv} = {l0v, . . . , lnv}, put civ := djv if li = ljv and civ = 0 if i 6∈ Iv. Then (apart from some modifications in the norms and the height) we can rewrite (1.1) as

A Generalization of the Subspace Theorem With Polynomials of Higher Degree

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 Open image in new window 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 Open image in new window [x 0,...,x n ]. Then the set of solutions x ∈ℙn(K) of

A further improvement of the Quantitative Subspace Theorem

\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)

General Analytical Solutions for Merton's-Type Consumption-Investment Problems

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

An effective version of Faltings' product theorem

1.1. Let S be a finite set of places of a finite field extension L of a number field K, containing all infinite places. Let E be a vector space over K of dimension N + 1. For v ∈ S, let lv0, · · · , lvN be linearly independent vectors in E ⊗K L. Choose a projective subvariety X defined over K, embedded into the projective space P(E∨) of lines of the dual vector space E∨. Consider the system of inequalities