Logarithms Of Algebraic Numbers Research Articles

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 .

On a Family of Quintic Thue Equations

We consider the parametrized quintic family of Thue equationsX(X2−Y2) (X2−a2Y2)−Y5=± 1 a∈Zand prove that it only has trivial solutions for ∣a∣≥ 3.6× 1019using a recent estimate for linear forms in three logarithms of algebraic numbers by P. Voutier.

Classification of hypergeometric identities for pi and other logarithms of algebraic numbers.

This paper provides transcendental and algebraic framework for the classification of identities expressing pi and other logarithms of algebraic numbers as rapidly convergent generalized hypergeometric series in rational parameters. Algebraic and arithmetic relations between values of p+1Fp hypergeometric functions and their values are analyzed. The existing identities are explained, and new exhaustive classes of new ones are presented.

Estimating Mahler's measure

In 1962, Mahler defined a measure for integer polynomials in several variables as the logarithmic integral over the torus. Many results exist about the values taken by the measure but many unsolved problems remain. In one variable, it is possible to express the measure as an effective limit of Riemann sums. We show that the same is true in several variables, using a non-obvious parametrisation of the torus together with Baker's Theorem on linear forms in logarithms of algebraic numbers.

Quadratic relations between logarithms of algebraic numbers

Quadratic relations between logarithms of algebraic numbers

Points whose coordinates are logarithms of algebraic numbers on algebraic varieties

Points whose coordinates are logarithms of algebraic numbers on algebraic varieties

A Small Contribution to Catalan′s Equation

Using recent results on linear forms in logarithms of algebraic numbers, we prove that any solution of the equation x p − y q = ϵ, where ϵ = ± 1, p and q are odd primes, and p > q satisfies p < 3.42 · 10 28 and q < 5.6 · 10 19. We also combine our work with some results of Altonen and Inkeri to determine the six cases with q ≤ 37 for which this equation may have solutions.

Nouvelles méthodes pour minorer des combinaisons linéaires de logarithmes de nombres algébriques

Up to now Baker’s method was the only one to yield lower bounds for linear forms in logarithms of algebraic numbers, at least when the number of logarithms is bigger than 2. We propose here other methods. While Baker’s work is a generalization of Gel’fond’s solution of Hilbert’s problem, our approach is based on Schneider’s solution of this problem. This first paper is devoted to the “dual” of a proof due to N. Hirata of a lower bound for linear forms in a commutative algebraic group (here we consider only the usual logarithms). In a subsequent paper we shall develop the “dual” of Baker’s method.

Constants for lower bounds for linear forms in the logarithms of algebraic numbers I. The general case

Constants for lower bounds for linear forms in the logarithms of algebraic numbers I. The general case

On the computation of all imaginary quadratic fields of class number one

Let d be the discriminant of an imaginary quadratic field with class number one. If d ⩽ − 10 4 d \leqslant - {10^4} it is easy to show, using an idea from Stark, that h ( 12 d ) ⩽ 2 | d | h(12d) \leqslant 2\sqrt {|d|} , h ( 24 d ) ⩽ 2 | d | h(24d) \leqslant 2\sqrt {|d|} and | h ( 24 d ) ln ⁡ ( 5 + 2 6 ) − 2 h ( 12 d ) ln ⁡ ( 2 + 3 ) | &gt; 50 exp ⁡ ( − π / 24 ⋅ | d | ) |h(24d)\ln (5 + 2\sqrt 6 ) - 2h(12d)\ln (2 + \sqrt 3 )| &gt; 50\exp ( - \pi /24 \cdot \sqrt {|d|} ) . This linear form is estimated for large | d | |d| from below with the aid of the quantitative version of Schneider’s α β {\alpha ^\beta } -theorem by Mignotte and Waldschmidt. In the "medium large" region 2 ⋅ 10 4 ⩽ | d | ⩽ 10 34 2 \cdot {10^4} \leqslant |d| \leqslant {10^{34}} it is shown by computing the beginning of the continued fraction of ln ⁡ ( 5 + 2 6 ) / ln ⁡ ( 2 + 3 ) \ln (5 + 2\sqrt 6 )/\ln (2 + \sqrt 3 ) that the above relations cannot hold.

Linear forms in elliptic logarithms

New lower bounds for linear forms in n (≥ 2) elliptic logarithms in the CM case are established. The estimate is better than all previous estimates with respect to some of the parameters that appear. It may be interesting to notice that the product log A 1 … log A n in the lower bound (see the Corollary of Theorem 1) is of exactly the same form as in the lower bounds for linear forms in logarithms of algebraic numbers (see A. Baker [ in “Transcendence Theory: Advances and Applications”. (A. Baker and D. W. Masser, Eds.), pp. 1–27, Academic Press, New York, 1977] ) and this is the first time such a parallelism has been achieved. To obtain the above lower bounds a zero estimate on the group variety G a n × E ( C n × E) is established (with E being an elliptic curve with CM), which is sharper than that derived from the general results in D. W. Masser and G. Wüstholz ( Inventiones Math. 63 (1981), 81–95).

The Hodge Theory of Flat Vector Bundles on a Complex Torus

We study the Hodge spectral sequence of a local system on a compact, complex torus by means of the theory of harmonic integrals. It is shown that, in some cases, Baker’s theorems concerning linear forms in the logarithms of algebraic numbers may be applied to obtain vanishing theorems in cohomology. This is applied to the study of Betti and Hodge numbers of compact analytic threefolds which are analogues of hyperelliptic surfaces. Among other things, it is shown that, in contrast to the two-dimensional case, some of these varieties are nonalgebraic.

The Hodge theory of flat vector bundles on a complex torus

We study the Hodge spectral sequence of a local system on a compact, complex torus by means of the theory of harmonic integrals. It is shown that, in some cases, Baker’s theorems concerning linear forms in the logarithms of algebraic numbers may be applied to obtain vanishing theorems in cohomology. This is applied to the study of Betti and Hodge numbers of compact analytic threefolds which are analogues of hyperelliptic surfaces. Among other things, it is shown that, in contrast to the two-dimensional case, some of these varieties are nonalgebraic.

Transcendence measures for exponentials and logarithms

AbstractIn the present paper, we derive transcendence measures for the numbers log α,eβ, αβ, (log α1)/(log α2) from a previous lower bound of ours on linear forms in the logarithms of algebraic numbers.

Multiplicative relations in number fields

In this paper, we obtain an explicit form of the currently best known inequality for linear forms in the logarithms of algebraic numbers. The results complete our previous investigations (Bull. Austral. Math. Soc. 15 (1976), 33–57) which were conditional on a certain independence condition on the algebraic numbers. The extra work needed to obtain unconditional results centres on the properties of multiplicative relations in number fields. In particular, we show that a set of multiplicatively dependent algebraic numbers always satisfies a relation with relatively small exponents.

On linear forms in the logarithms of algebraic numbers

On linear forms in the logarithms of algebraic numbers

On a problem of Chowla

The equation Σ f(n) n = 0 is studied for periodic algebraically-valued functions f and, in particular, a well known problem of Chowla in this context is resolved. The work depends on an application of a theorem of the first author concerning linear forms in the logarithms of algebraic numbers.

Imaginary quadratic fields of class number 2

A conjecture concerning linear forms in the logarithms of algebraic numbers is made. It is shown that this conjecture allows an effective determination of all imaginary quadratic fields of class number 2.

On Stark's theorem

A new proof is obtained for Gauss' hypothesis on class-one imaginary quadratic fields, based on the bound obtained by A. O. Gel'fond in 1939 for the modulus of linear forms of two logarithms of algebraic numbers.