arXiv subjects
Publications and source records attributed to Jan-Hendrik Evertse.
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).
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.
Given a fixed number field K, we give non-trivial lower bounds for the distance between the conjugates of any number a from K in terms of the Mahler measure M(a) of a. In the case that K is a cubic field, our bound is best possible in terms of M(a). Our proof is based on Roth's theorem.
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.
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.
We give an explicit upper bound for the number of equivalence classes of binary forms with rational integral coefficients of given degree and given discriminant, and with given splitting field. Further, we give an explicit upper bound for the number of irreducible binary forms with rational integral coefficients with given invariant order. Our bounds depend on as few parameters as possible. For instance, we show that the number of equivalence classes of irreducible binary forms with rational integral coefficients of degree r with given invariant order has an upper bound depending only on r. We have proved more general results for binary forms with coefficients in the ring of S-integers of a number field.