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20
Ternary form equations
 J. Number Theory
, 1995
"... Let T be a homogeneous polynomial in three variables and with rational integer coëfficients. As a generalisation of Thue’s equation we consider the ternary form equation T (x, y, z) = 1 in the integral unknowns x, y, z. We prove some general results when the degree of T is at most three and make so ..."
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Let T be a homogeneous polynomial in three variables and with rational integer coëfficients. As a generalisation of Thue’s equation we consider the ternary form equation T (x, y, z) = 1 in the integral unknowns x, y, z. We prove some general results when the degree of T is at most three and make some modest inroads into the case degT> 3. Generalisations to algebraic number fields are considered at the same time. 1
Exceptional Units and Numbers of Small Mahler Measure
 Math
"... this article we investigate how many powers of ff can be exceptional units. Thus we will be looking at solutions of the unit equation ..."
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this article we investigate how many powers of ff can be exceptional units. Thus we will be looking at solutions of the unit equation
Multivariate Diophantine equations with many solutions
, 2001
"... Among other things we show that for each ntuple of positive rational numbers (a 1 ; : : : ; a n ) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a 1 x 1 + +a n x n = 1 with x 1 ; : : : ; x n Sunits are not contained in fewer than exp((4 + ..."
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Among other things we show that for each ntuple of positive rational numbers (a 1 ; : : : ; a n ) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a 1 x 1 + +a n x n = 1 with x 1 ; : : : ; x n Sunits are not contained in fewer than exp((4 + o(1))s 1=2 (log s) 1=2 ) proper linear subspaces of C n . This generalizes a result of Erdős, Stewart and Tijdeman [7] for Sunit equations in two variables. Further, we prove that for any algebraic number field K of degree n, any integer m with 1 m < n, and any sufficiently large s there are integers 0 ; : : : ; m in K which are linearly independent over Q , and prime numbers p 1 ; : : : ; p s , such that the norm polynomial equation jN K=Q ( 0 + 1 x 1 + + mxm )j = p z1 1 p zs s has at least expf(1+o(1)) n m s m=n (log s) 1+m=n g solutions in x 1 ; : : : ; xm ; z 1 ; : : : ; z s 2 Z. This generalizes a result of Moree and Stewart [19] for m = 1. Our main tool, also established in this paper, is an effective lower bound for the number K;T (X; Y ) of ideals in a number field K of norm X composed of prime ideals which lie outside a given finite set of prime ideals T and which have norm Y . This generalizes results of Caneld, Erdős and Pomerance [6] and of Moree and Stewart [19].
POWERS IN FINITELY GENERATED GROUPS
"... Abstract. In this paper we study the set Γn of nthpowers in certain finitely generated groups Γ. We show that, if Γ is soluble or linear, and Γn contains a finite index subgroup, then Γ is nilpotentbyfinite. We also show that, if Γ is linear and Γn has finite index (i.e. Γ may be covered by finit ..."
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Abstract. In this paper we study the set Γn of nthpowers in certain finitely generated groups Γ. We show that, if Γ is soluble or linear, and Γn contains a finite index subgroup, then Γ is nilpotentbyfinite. We also show that, if Γ is linear and Γn has finite index (i.e. Γ may be covered by finitely many translations of Γn), then Γ is solublebyfinite. The proof applies invariant measures on amenable groups, numbertheoretic results concerning the Sunit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic. 1.
Two Sunit equations with many solutions
 J. Number Theory
, 2007
"... In this note we consider two Sunit equations for which we will exhibit many solutions. Our first problem concerns solutions to the equation a + b = c where a, b, and c are coprime integers such that all prime factors of abc lie in a given set S of s primes. In [8] J.H. Evertse showed that this Su ..."
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In this note we consider two Sunit equations for which we will exhibit many solutions. Our first problem concerns solutions to the equation a + b = c where a, b, and c are coprime integers such that all prime factors of abc lie in a given set S of s primes. In [8] J.H. Evertse showed that this Sunit equation has at most exp(4s + 6) solutions. On the other
A GENERALIZATION OF THE SUBSPACE THEOREM WITH POLYNOMIALS OF HIGHER DEGREE
, 2004
"... Abstract. Recently, Corvaja and Zannier [2, Theorem 3] proved an extension of the Subspace Theorem with polynomials of arbitrary degree 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 this ..."
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Abstract. Recently, Corvaja and Zannier [2, Theorem 3] proved an extension of the Subspace Theorem with polynomials of arbitrary degree 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 this 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 in a finite number of proper subvarieties of X, with explicit upper bounds for the number and for the degrees of these subvarieties (Theorem 1.3 below). We deduce our generalization from a general result on twisted heights on projective varieties (Theorem 2.1 in Section 2). Our main tools are the quantitative version of the Absolute Parametric Subspace Theorem by Evertse and Schlickewei [5, Theorem 1.2], as well as a lower bound by Evertse and Ferretti [4, Theorem 4.1] for the normalized Chow weight of a projective variety in terms of its mth normalized Hilbert weight. 1.
The Number Of Solutions Of Diophantine Equations
"... Introduction. In two recent papers [4], [30], Erdos, Stewart and the author showed that certain diophantine equations have many solutions. In this way they indicated how far certain results are capable for improvements at most. First we mention some relevant results from the literature on upper boun ..."
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Introduction. In two recent papers [4], [30], Erdos, Stewart and the author showed that certain diophantine equations have many solutions. In this way they indicated how far certain results are capable for improvements at most. First we mention some relevant results from the literature on upper bounds for the numbers of solutions of diophantine equations and then we sketch how our method leads to opposite results. 1. Thue and ThueMahler equations. Let f(x; y) = a 0 x n + a 1 x n\Gamma1 y + ::: + an y n 2 Z [x ; y ] be a binary form (i.e. homogeneous polynomial) of degree n 3: Put A := max j=0;:::;n ja j j. Let m<F39.