My main research interests are in Number theory
(2000 Mathematics Subject Classification 11),
in particular in Diophantine approximation (11J) and applications of the
latter to Diophantine equations (11D) and inequalities (11J).
My recent research concerns
S-unit equations over function fields,
generalizations of Schmidt's Subspace Theorem,
estimating the number of equivalence classes of binary forms
with given discriminant, and finding better lower bounds for the
distances between the conjugates of an algebraic number.
Originally, Diophantine approximation deals with approximation of real numbers
by rational numbers and with questions whether certain given numbers are irrational,
algebraic, or transcendental. The basic techniques from Diophantine
approximation have been vastly generalized, and there have been some striking
applications to Diophantine equations and inequalities and to arithmetic
algebraic geometry (11G, 14G). For instance, by means of Diophantine aproximation
techniques, in 1990, G. Faltings proved the following general conjecture of S. Lang:
let A be an abelian variety defined over an algebraic number field
K and let X be an algebraic subvariety of A, also
defined over K, which does not contain a translate of an abelian subvariety
of A. Then X has only finitely many K-rational points.
This is a higher dimensional generalization of the Mordell conjecture, proved by Faltings in 1983: if C is a projective curve of genus at least 2 defined
over a number field K, then C has only finitely many
K-rational points.
Much of my research has been related to the Subspace Theorem,
proved by W.M. Schmidt in 1972:
let L_{1}(x)
=a_{11}x_{1}+...+
a_{1n}x_{n}
, . . . ,
L_{n}(x)
=a_{n1}x_{1}+...+
a_{nn}x_{n}
be linearly independent linear forms with algebraic coefficients.
Let d>0. Then the set of points
x=(x_{1},...,x_{n})
in Z^{n} satisfying the inequality
(*)
|L_{1}(x)...
L_{n}(x)| <
(max (|x_{1}|,...,
|x_{n}|)^{-d}
is contained in the union of finitely many proper linear subspaces of
Q^{n}.
Schmidt's proof uses Diophantine approximation techniques as well as techniques
from the geometry of numbers. In 1994,
Faltings and Wüstholz gave a totally different
proof which avoids the geometry of numbers but which uses instead
Faltings' Product Theorem. A notorious drawback of both the
proofs of Schmidt and of Faltings and Wüstholz is, that they are
ineffective, i.e., they do not provide an effective algorithm
to find the subspaces.
The Subspace Theorem is a higher dimensional generalization of Roth's theorem
from 1955: for each algebraic number a and for each k> 2,
there are only finitely many pairs of integers (p,q) with q>0
such that |a-(p/q)|<q^{-k
}.
The Subspace Theorem has been extended and refined in several
directions (p-adic versions, inequalities with solutions from a number field
(Schmidt, Schlickewei),
quantitative versions giving explicit upper bounds for the number of subspaces
(Schmidt, Schlickewei, E.)).
In another direction, Vojta showed that there is a finite collection of proper
linear subspaces of Q^{n}, which is effectively
computable and is independent of d, such that all but finitely many solutions
of (*) lie in the union of these subspaces.
Further, Ru and Vojta proved a "moving targets version" of the Subspace
Theorem, yielding a finiteness result in a situation where also the
linear forms L_{1},...,
L_{n} (the 'targets') are allowed to vary,
though in a small range.
The Subspace Theorem and its generalizations and extensions have numerous
applications, for instance to
Diophantine inequalities (simultaneous approximation of algebraic numbers by
rationals (Schmidt), approximation of algebraic numbers by algebraic numbers of bounded degree (Wirsing, Schmidt),
"symmetric" approximation results implying that there are only finitely many
pairs of algebraic numbers (a,b) in a given number field that are close
together (E.), inequalities for resultants of two polynomials (E.)),
Diophantine equations (finiteness results for S-unit equations
(Schlickewei, Dubois and Rhin, van der Poorten and Schlickewei, E.),
explicit upper bounds for the number of
solutions of linear equations with unknowns from a multiplicative group of
finite rank (Schlickewei, Schmidt, E.),
norm form equations, decomposable form equations
(Schmidt, Györy, Gaál, E., Ru, Berczés, Everest, Thunder),
linear recurrence sequences (van der Poorten, Schlickewei, Schmidt, Corvaja, Zannier),
exponential polynomial equations (Laurent, Schlickewei, Schmidt),
transcendence results, e.g., for Mahler-type gap series (Nishioka, Corvaja, Zannier, Bugeaud),
integral points on varieties (Corvaja, Zannier, Levin, Autissier, Vojta),
and complexity of b-ary expansions of algebraic numbers and of continued fraction expansions of algebraic numbers (Adamczewski, Bugeaud).
A further development
is to generalize
the theory related to the Subspace Theorem to
inequalities to be solved in algebraic points
of a projective variety. Faltings and Wüstholz pointed out that their method
to attack the Subspace Theorem can be used also to handle inequalities of the
shape (*) in which the L_{i} are homogeneous
polynomials of arbitrary degree instead of just linear forms, and moreover,
the solutions x are chosen from
an arbitrary projective variety.
Ferretti and E. showed that this generalization of Faltings and Wüstholz
can be deduced from Schmidt's Subspace Theorem and moreover, using the
quantitative version of Schmidt's Subspace Theorem of Schlickewei and E.,
they obtained a quantitative version of the result of Faltings and
Wüstholz. Similar such work has been done by Corvaja and Zannier.
In the survey
Diophantine equations and Diophantine approximation
which is (supposedly)
meant for non-specialists
I have sketched some of the consequences concerning
linear equations with unknowns from a multiplicative group of
finite rank and linear recurrence sequences.