Logarithms Of Algebraic Numbers Research Articles

Minorations simultanées de formes linéaires de logarithmes de nombres algébriques

This work falls within the theory of linear forms in logarithms over a commutative linear group defined over a number field. We give lower bounds for simultaneous linear forms in logarithms of algebraic numbers, treating both the archimedean and $p$-adic cases. The proof includes Baker's method, Hirata's reduction, Chudnovsky's process of variable change. The novelty is that we integrated into the proof the modern tools of adelic slope theory, using also a new small values Siegel's lemma.

On the number of solutions of binomial Thue inequalities

Abstract Let a , b and n be positive integers with n ⩾ 3 and consider the binomial Thue inequality | a x n − b y n | ⩽ 3 . In this paper, we extend a result of the first author and prove that, apart from finitely many explicitly given exceptions, this inequality has at most a single solution in positive integers x and y . In the proof, we combine lower bounds for linear forms in logarithms of algebraic numbers with the hypergeometric method of Thue-Siegel and an assortment of techniques from computational Diophantine approximation.

Multiplicity estimates and degeneration

In this paper we prove a multiplicity estimate, which is best possible up to a multiplicative constant, in which the set of points is connected to an action of SL2(Z) on the torus \({{\bf G}_m^2({\bf C})}\). This result is motivated by the construction, due to Roy, of a non-trivial auxiliary function that could be used to study the points on the Grassmannian whose coordinates are logarithms of algebraic numbers: to make use of this construction, only a zero estimate connected to the action of GL m (Z) on \({\wedge^k {\bf C}^m}\) is missing. The result we prove is essentially analogous to it. The proof is based on the fact that a zero (or multiplicity) estimate can be derived from a lower bound for a Seshadri constant. Then a degeneration argument is used: inside a family any such lower bound holds on an open subset, so proving it for sufficiently many special cases yields it for almost all cases.

On the largest prime factor of the partition function of n

Let p(n) be the function that counts the number of partitions of n. For a positive integer m, let P(m) be the largest prime factor of m. Here, we show that P(p(n)) tends to infinity when n tends to infinity through some set of asymptotic density 1. In fact, we show that the inequality P(p(n))>loglogloglogloglogn holds for almost all positive integers n. Features of the proof are the first term in Rademacher’s formula for p(n), Gowers’ effective version of Szemeredi’s theorem, and a classical lower bound for a nonzero homogeneous linear form in logarithms of algebraic numbers due to Matveev.

Digit sums of binomial sums

Let b ⩾ 2 be a fixed positive integer and let S ( n ) be a certain type of binomial sum. In this paper, we show that for most n the sum of the digits of S ( n ) in base b is at least c 0 log n / ( log log n ) , where c 0 is some positive constant depending on b and on the sequence of binomial sums. Our results include middle binomial coefficients ( 2 n n ) and Apéry numbers A n . The proof uses a result of McIntosh regarding the asymptotic expansions of such binomial sums as well as Bakerʼs theorem on lower bounds for nonzero linear forms in logarithms of algebraic numbers.

Linear forms at a basis of an algebraic number field

It was proved by Cassels and Swinnerton-Dyer that the Littlewood conjecture in simultaneous Diophantine approximation holds for any pair of numbers in a cubic field. Later this result was generalized by Peck to a basis ( 1 , α 1 , … , α n ) of a real algebraic number field of degree at least 3. By transference, this result provides some solutions for the dual form of Littlewoodʼs conjecture. Here we find another solutions, and using Bakerʼs estimates for linear forms in logarithms of algebraic numbers, we discuss whether the result is best possible.

Determination of integer solutions of a system of simultaneous Pell equations

We describe an algorithm for finding integer solutions of a system of simultaneous Pell equations whose effective estimates can be obtained using the theory of linear forms in the logarithms of algebraic numbers. We use Matveev’s estimate for forms in three logarithms. To decrease the resulting estimate, we use an iterative algorithm. At the end of the paper, results of the practical implementation of the proposed algorithm are given.

The Hilbert polynomial and linear forms in the logarithms of algebraic numbers

We prove a new estimate for homogeneous linear forms with integer coefficients in the logarithms of algebraic numbers. We obtain a qualitative improvement of the estimate depending on the coefficients of the linear form and the best value of the constant in the estimate in the case when the number of logarithms is not too large.

Diophantische Approximationen

Diophantische Approximationen

Index of lattices and Hilbert polynomials

An upper bound for the index of a sublattice, which arises in relation to various versions of zero lemmas in the theory of linear forms in logarithms of algebraic numbers, in terms of the Hilbert polynomial is found. Simultaneously, a lower bound for the values of this polynomial is obtained.

The index of multiplicative groups of algebraic numbers

Fix several non-zero elements of an algebraic field with linearly independent logarithms. Consider the set of elements of the field whose logarithms can be expressed in terms of the logarithms of the fixed numbers using rational coefficients. The corresponding vectors of coefficients make up a lattice with the standard integral lattice as a finite-index sublattice. An improved upper bound for this index is given in terms of the extended logarithmic heights of the quantities involved. On the way an estimate for the coefficients in integer linear relations between the logarithms of algebraic numbers is obtained.

Further Variations on the Six Exponentials Theorem.

According to the Six Exponentials Theorem, a $2\times 3$ matrix whose entries $\lambda_{ij}$ ($i=1,2$, $j=1,2,3$) are logarithms of algebraic numbers has rank $2$, as soon as the two rows as well as the three columns are linearly independent over the field $\BbbQ$ of rational numbers. The main result of the present note is that one at least of the three $2\times 2$ determinants, viz. $$ \lambda_{21}\lambda_{12}-\lambda_{11}\lambda_{22}, \quad \lambda_{22}\lambda_{13}-\lambda_{12}\lambda_{23}, \quad \lambda_{23}\lambda_{11}-\lambda_{13}\lambda_{21} $$ is transcendental.

Pillai's conjecture revisited

We prove a generalization of an old conjecture of Pillai (now a theorem of Stroeker and Tijdeman) to the effect that the Diophantine equation 3x−2y=c has, for |c|>13, at most one solution in positive integers x and y. In fact, we show that if N and c are positive integers with N⩾2, then the equation |(N+1)x−Ny|=c has at most one solution in positive integers x and y, unless (N,c)∈{(2,1),(2,5),(2,7),(2,13),(2,23),(3,13)}. Our proof uses the hypergeometric method of Thue and Siegel and avoids application of lower bounds for linear forms in logarithms of algebraic numbers.

On a conjecture of E. Thomas concerning parametrized Thue equations

is called Thue equation in honour of A. Thue, who proved in 1909 [19] that the number of its solutions in integers is finite. Thue’s result is not effective, but in 1968, A. Baker [1] gave an upper bound for the solutions using his lower bounds for linear forms in logarithms of algebraic numbers. Since then, algorithms for the solution of single Thue equations have been ∗This work was supported by Austrian National Bank project nr. 7088 and the Austrian-Hungarian Science Cooperation project

Orbits under algebraic groups and logarithms of algebraic numbers

Orbits under algebraic groups and logarithms of algebraic numbers

An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II

In this paper we establish an estimate for the modulus of a linear form of the type described in the title. This estimate is exponential with respect to the number of summands.

On the representation of unity by binary cubic forms

If F (x, y) is a binary cubic form with integer coefficients such that F (x, 1) has at least two distinct complex roots, then the equation F (x, y) = 1 possesses at most ten solutions in integers x and y, nine if F has a nontrivial automorphism group. If, further, F (x, y) is reducible over Z[x, y], then this equation has at most 2 solutions, unless F (x, y) is equivalent under GL2(Z)action to either x(x2 − xy − y2) or x(x2 − 2y2). The proofs of these results rely upon the method of Thue-Siegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations F (x, y) = 1 for F cubic and irreducible of positive discriminant DF ≤ 106. As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form F (x, y) = m and to Mordell’s equation y2 = x3 + k, where m and k are nonzero integers.

The Diophantine equation $b^2X^4-dY^2=1$

If $b$ and $d$ are given positive integers with $b > 1$, then we show that the equation of the title possesses at most one solution in positive integers $X,Y$. Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic fields. The proof utilizes estimates for linear forms in logarithms of algebraic numbers in conjunction with properties of Pellian equations and the Jacobi symbol and explicit determination of the integer points on certain elliptic curves.

Effective solution of families of Thue equations containing several parameters

F (X,Y ) = m, where F ∈ Z[X,Y ] is an irreducible form of degree n ≥ 3 and m 6= 0 a fixed integer, has only finitely many solutions. However, this proof is non-effective and does not give any bounds for the size of the possible solutions. In 1968, A. Baker could give effective bounds based on his famous theory on linear forms in logarithms of algebraic numbers. In the last decades, this method was refined (see for instance Baker and Wustholz [1] and Waldschmidt [29]). Recent explicit upper bounds for the solutions of Thue equations have been given by Bugeaud and Győry [3]. Algorithms for the solution of a single Thue equation have been developed by several authors (see Bilu and Hanrot [2]). E. Thomas [25] was the first to deal with a parametrized family of Thue equations; since then, some families have been investigated: cubic families have been discussed by Mignotte [16], Lee [12], and Mignotte and Tzanakis [19], a cubic inequality has been solved by Mignotte, Pethő, and Lemmermeyer [17]; quartic families have been considered by Pethő [20], Mignotte, Pethő, and Roth [18], Lettl and Pethő [13], Chen and Voutier [4], and Heuberger, Pethő, and Tichy [10]; Wakabayashi [28] dealt with a quartic inequality, Pethő and Tichy [21] solved a two-parametric quartic family, quintic families have been investigated by Heuberger [9] and Gaal and Lettl [6], a sextic family has been solved by Lettl, Pethő, and Voutier [14, 15]; see

An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers

In this paper we study linear forms with rational integer coefficients (, ), where the are algebraic numbers satisfying the so-called strong independence condition. In standard notation, we prove an explicit estimate of the form Its novel feature is that it contains no factors of the form .