78 research outputs found
On the Quantitative Subspace Theorem
In this survey we give an overview of recent developments on the Quantitative
Subspace Theorem. In particular, we discuss a new upper bound for the number of
subspaces containing the "large" solutions, obtained jointly with Roberto
Ferretti, and sketch the proof of the latter. Further, we prove a new gap
principle to handle the "small" solutions in the system of inequalities
considered in the Subspace Theorem. Finally, we go into the refinement of the
Subspace Theorem by Faltings and Wuestholz, which states that the system of
inequalities considered has only finitely many solutions outside some
effectively determinable proper linear subspace of the ambient solution space.
Estimating the number of these solutions is still an open problem. We give some
motivation that this problem is very hard.Comment: 26 page
Mahler's work on the geometry of numbers
Mahler has written many papers on the geometry of numbers. Arguably, his most
influential achievements in this area are his compactness theorem for lattices,
his work on star bodies and their critical lattices, and his estimates for the
successive minima of reciprocal convex bodies and compound convex bodies. We
give a, by far not complete, overview of Mahler's work on these topics and
their impact.Comment: 17 pages. This paper will appear in "Mahler Selecta", a volume
dedicated to the work of Kurt Mahler and its impac
Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces
Let K be a field of characteristic 0. We consider linear equations
a1*x1+...+an*xn=1 in unknowns x1,...,xn from G, where a1,...,an are non-zero
elements of K, and where G is a subgroup of the multiplicative group of
non-zero elements of K. Two tuples (a1,...,an) and (b1,...,bn) of non-zero
elements of K are called G-equivalent if there are u1,...,un in G such that
b1=a1*u1,..., bn=an*un. Denote by m(a1,...,an,G) the smallest number m such
that the set of solutions of a1*x1+...+an*xn=1 in x1,...,xn from G is contained
in the union of m proper linear subspaces of K^n. It is known that
m(a1,...,an,G) is finite; clearly, this quantity does not change if (a1,...,an)
is replaced by a G-equivalent tuple. Gyory and the author proved in 1988 that
there is a constant c(n) depending only on the number of variables n, such that
for all but finitely many G-equivalence classes (a1,...,an), one has
m(a1,...,an,G)< c(n). It is as yet not clear what is the best possible value of
c(n). Gyory and the author showed that c(n)=2^{(n+1)!} can be taken. This was
improved by the author in 1993 to c(n)=(n!)^{2n+2}. In the present paper we
improve this further to c(n)=2^{n+1}, and give an example showing that c(n) can
not be smaller than n.Comment: 12 pages, latex fil
On two notions of complexity of algebraic numbers
we derive new, improved lower bounds for the block complexity of an
irrational algebraic number and for the number of digit changes in the b-ary
expansion of an irrational algebraic number. To this end, we apply a
quantitative version of the Subspace Theorem due to Evertse and Schlickewei
(2002).Comment: 31 page
Approximation of complex algebraic numbers by algebraic numbers of bounded degree
We investigate how well complex algebraic numbers can be approximated by
algebraic numbers of degree at most n. We also investigate how well complex
algebraic numbers can be approximated by algebraic integers of degree at most
n+1. It follows from our investigations that for every positive integer n there
are complex algebraic numbers of degree larger than n that are better
approximable by algebraic numbers of degree at most n than almost all complex
numbers. As it turns out, these numbers are more badly approximable by
algebraic integers of degree at most n+1 than almost all complex numbers.Comment: 34 page
Effective results for discriminant equations over finitely generated domains
Let be an integral domain with quotient field of characteristic
that is finitely generated as a -algebra. Denote by the
discriminant of a polynomial . Further, given a finite etale algebra
, we denote by the discriminant of
over . For non-zero , we consider equations
to be solved in monic polynomials of given degree having
their zeros in a given finite extension field of , and
D_{\Omega/K}(\alpha)=\delta\,\,\mbox{ in } \alpha\in O, where is an
-order of , i.e., a subring of the integral closure of in
that contains as well as a -basis of .
In our book ``Discriminant Equations in Diophantine Number Theory, which will
be published by Cambridge University Press we proved that if is effectively
given in a well-defined sense and integrally closed, then up to natural notions
of equivalence the above equations have only finitely many solutions, and that
moreover, a full system of representatives for the equivalence classes can be
determined effectively. In the present paper, we extend these results to
integral domains that are not necessarily integrally closed.Comment: 20 page
Linear equations with unknowns from a multiplicative group in a function field
Let k be an algebraically closed field of characteristic 0, let K/k be a
transcendental extension of arbitrary transcendence degree and let G be a
multiplicative subgroup of (K^*)^n such that (k^*)^n is contained in G, and
G/(k^*)^n has finite rank r. We consider linear equations a1x1+...+anxn=1 (*)
with fixed non-zero coefficients a1,...,an from K, and with unknowns
(x1,...,xn) from the group G. Such a solution is called degenerate if there is
a subset of a1x1,...,anxn whose sum equals 0. Two solutions (x1,...,xn),
(y1,...,yn) are said to belong to the same (k^*)^n-coset if there are c1,...,cn
in k^* such that y1=c1*x1,...,yn=cn*xn. We show that the non-degenerate
solutions of (*) lie in at most 1+C(3,2)^r+C(4,2)^r+...+C(n+1,2)^r
(k^*)^n-cosets, where C(a,b) denotes the binomial coefficient a choose b.Comment: 15 pages, LaTeX fil
Effective results for unit equations over finitely generated domains
Let A be a commutative domain containing Z which is finitely generated as a
Z-algebra, and let a,b,c be non-zero elements of A. It follows from work of
Siegel, Mahler, Parry and Lang that the equation (*) ax+by=c has only finitely
many solutions in elements x,y of the unit group A* of A, but the proof
following from their arguments is ineffective. Using linear forms in logarithms
estimates of Baker and Coates, in 1979 Gy\H{o}ry gave an effective proof of
this finiteness result, in the special case that A is the ring of S-integers of
an algebraic number field. Some years later, Gy\H{o}ry extended this to a
restricted class of finitely generated domains A, containing transcendental
elements. In the present paper, we give an effective finiteness proof for the
number of solutions of (*) for arbitrary domains A finitely generated over Z.
In fact, we give an explicit upper bound for the `sizes' of the solutions x,y,
in terms of defining parameters for A,a,b,c. In our proof, we use already
existing effective finiteness results for two variable S-unit equations over
number fields due to Gy\H{o}ry and Yu and over function fields due to Mason, as
well as an explicit specialization argument.Comment: 41 page
A generalization of the Subspace Theorem with polynomials of higher degree
Recently, Corvaja and Zannier obtained an extension of the Subspace Theorem
with arbitrary homogeneous polynomials of arbitrary degreee instead of linear
forms. Their result states that the set of solutions in P^n(K) (K number field)
of the inequality being considered is not Zariski dense. In our paper we prove
by a different method a generalization of their result, in which the solutions
are taken from an arbitrary projective variety X instead of P^n. Further, we
give a quantitative version which states in a precise form that the solutions
with large height lie ina finite number of proper subvarieties of X, with
explicit upper bounds for the number and for the degrees of these subvarieties.Comment: 31 page
Orders with few rational monogenizations
For an algebraic number of degree , let be
the -module generated by ; then
is the ring of scalars
of . We call an order of the shape
\emph{rationally monogenic}. If is an algebraic integer, then
is monogenic. Rationally monogenic
orders are special types of invariant orders of binary forms, which have been
studied intensively. If are two
-equivalent algebraic numbers, i.e., for some
,
then . Given an order of
a number field, we call a -equivalence class of
with a \emph{rational
monogenization} of .
We prove the following. If is a quartic number field, then has only
finitely many orders with more than two rational monogenizations. This is best
possible. Further, if is a number field of degree , the Galois
group of whose normal closure is -transitive, then has only finitely
many orders with more than one rational monogenization. The proof uses
finiteness results for unit equations, which in turn were derived from
Schmidt's Subspace Theorem. We generalize the above results to rationally
monogenic orders over rings of -integers of number fields. Our results
extend work of B\'{e}rczes, Gy\H{o}ry and the author from 2013 on multiply
monogenic orders.Comment: This is the final version which has been published on-line by Acta
Arithmetica. It is a slight modification of the previous versio
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