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

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

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

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

Effective results for discriminant equations over finitely generated domains

Let $A$ be an integral domain with quotient field $K$ of characteristic $0$ that is finitely generated as a $\mathbb{Z}$-algebra. Denote by $D(F)$ the discriminant of a polynomial $F\in A[X]$. Further, given a finite etale algebra $\Omega$, we denote by $D_{\Omega/K}(\alpha )$ the discriminant of $\alpha$ over $K$. For non-zero $\delta\in A$, we consider equations $$D(F)=\delta$$ to be solved in monic polynomials $F\in A[X]$ of given degree $n\geq 2$ having their zeros in a given finite extension field $G$ of $K$, and D_{\Omega/K}(\alpha)=\delta\,\,\mbox{ in } \alpha\in O, where $O$ is an $A$-order of $\Omega$, i.e., a subring of the integral closure of $A$ in $\Omega$ that contains $A$ as well as a $K$-basis of $\Omega$. In our book ``Discriminant Equations in Diophantine Number Theory, which will be published by Cambridge University Press we proved that if $A$ 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 $A$ that are not necessarily integrally closed.Comment: 20 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 $\alpha$ of degree $n$, let $\mathcal{M}_{\alpha}$ be the $\mathbb{Z}$-module generated by $1,\alpha ,\ldots ,\alpha^{n-1}$; then $\mathbb{Z}_{\alpha}:=\{\xi\in\mathbb{Q} (\alpha ):\, \xi\mathcal{M}_{\alpha}\subseteq\mathcal{M}_{\alpha}\}$ is the ring of scalars of $\mathcal{M}_{\alpha}$. We call an order of the shape $\mathbb{Z}_{\alpha}$ \emph{rationally monogenic}. If $\alpha$ is an algebraic integer, then $\mathbb{Z}_{\alpha}=\mathbb{Z}[\alpha ]$ is monogenic. Rationally monogenic orders are special types of invariant orders of binary forms, which have been studied intensively. If $\alpha ,\beta$ are two $\text{GL}_2(\mathbb{Z})$-equivalent algebraic numbers, i.e., $\beta =(a\alpha +b)/(c\alpha +d)$ for some $\big(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\big)\in\text{GL}_2(\mathbb{Z})$, then $\mathbb{Z}_{\alpha}=\mathbb{Z}_{\beta}$. Given an order $\mathcal{O}$ of a number field, we call a $\text{GL}_2(\mathbb{Z})$-equivalence class of $\alpha$ with $\mathbb{Z}_{\alpha}=\mathcal{O}$ a \emph{rational monogenization} of $\mathcal{O}$. We prove the following. If $K$ is a quartic number field, then $K$ has only finitely many orders with more than two rational monogenizations. This is best possible. Further, if $K$ is a number field of degree $\geq 5$, the Galois group of whose normal closure is $5$-transitive, then $K$ 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 $S$-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