Linear forms in logarithms

Baker’s method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we generalize results of Levin regarding Baker’s method for varieties, and explain how, quite surprisingly, it mixes (under additional hypotheses) with Runge’s method to improve some known estimates in the case of curves by bypassing (or more generally reducing) the need for linear forms in p-adic logarithms. We then use these ideas to improve known estimates on solutions of S-unit equations. Finally, we explain how a finer analysis and formalism can improve upon the conditions given, and give some applications to the Siegel modular variety A2(2).

Linear forms in elliptic logarithms

This expository paper explores the theory of linear forms in logarithms and its applications to Diophantine equations. We begin with foundational results on transcendental numbers, including Liouville's theorem and the Gelfond-Schneider theorem, before developing Baker's theory of linear forms in logarithms. The paper concludes with applications to Diophantine equations through the Baker-Davenport method, illustrating these techniques with concrete examples. The purpose of this paper is to provide a step-by-step understanding of this particular area of Mathematics, and was written as a part of Euler Circle's Independent Paper and Research Writing Program, in which the author studied the topic independently and wrote this paper over a period of 4 weeks.

The history of the theory of linear forms in logarithms is well known. We shall briefly sketch only some of the moments connected with new technical progress and important for our article. This theory was originated by pioneer works of A.O. Gelfond (see, for example, [5, 6]); with the help of the ideas which arose in connection with the solution of 7-th Hilbert problem (construction of auxiliary functions, extrapolation of zeros and small values), the bounds for the homogenous linear forms in two logarithms were proved. In the middle of the sixties A. Baker, [1], using auxiliary functions in several complex variables, for the first time obtained bounds for linear forms in any number of logarithms, both in the homogenous and non-homogenous cases. All further development of the theory is connected with improvements of these bounds. So N.I. Feldman introduced the so-called binomial polynomials in the construction of the auxiliary function, and due to this the dependence of the bounds on the coefficients of the linear forms was improved. The Kummer theory was used by A. Baker and H. Stark, [2], to improve the dependence of the estimates on the parameters related to the algebraic numbers appearing as arguments of the logarithms. The further improvements of this dependence are connected with the introduction of bounds for the number of zeros of polynomials on algebraic groups in works of G. Wustholz. [14], P. Philippon and M. Waldschmidt, [11], A. Baker and G. Wustholz, [3].

The following two decision problems capture the complexity of comparing integers or rationals that are succinctly represented in product-of-exponentials notation, or equivalently, via arithmetic circuits using only multiplication and division gates, and integer inputs. Input instance: Four lists of positive integers: a 1,..., an ∈N+ n ; b 1,..., bn ∈N+ n ; c 1,..., cm ∈N+ m ; d 1, ..., dm ∈N+ m ; where each of the integers is represented in binary. Problem 1 (equality testing): Decide whether a 1 b 1 a 2 b 2⋯ anbn = c 1 d 1 c 2 d 2⋯ cmdm . Problem 2 (inequality testing): Decide whether a 1 b 1 a 2 b 2⋯ anbn ≥ c 1 d 1 c 2 d 2⋯ cmdm . Problem 1 is easily decidable in polynomial time using a simple iterative algorithm. Problem 2 is much harder. We observe that the complexity of Problem 2 is intimately connected to deep conjectures and results in number theory. In particular, if a refined form of the ABC conjecture formulated by Baker in 1998 holds, or if the older Lang-Waldschmidt conjecture (formulated in 1978) on linear forms in logarithms holds, then Problem 2 is decidable in P-time (in the standard Turing model of computation). Moreover, it follows from the best available quantitative bounds on linear forms in logarithms, namely, by Baker and Wüstholz [1993] or Matveev [2000], that if m and n are fixed universal constants then Problem 2 is decidable in P-time (without relying on any conjectures). This latter fact was observed earlier by Shub [1993]. We describe one application: P-time maximum probability parsing for arbitrary stochastic context-free grammars (where ε -rules are allowed).

This paper aims to show two things. Firstly the importance of Alan Baker's work on linear forms in logarithms for the development of the theory of exponential Diophantine equations. Secondly how this theory is the culmination of a series of greater and smaller discoveries.

Matrices whose coefficients are linear forms in logarithms

Using linear forms in logarithms we prove an explicit result of Andre Oort type for P-1(C) x G(m)(C). In this variation the special points of P-1(C) x G(m)(C) are of the form (alpha, lambda), with a a singular modulus and lambda a root of unity. The qualitative version of our result states that if C is a closed algebraic curve in P-1(C) x G(m)(C), defined over a number field, not containing a horizontal or vertical line, then C contains only finitely many special points. The proof is based on linear forms in logarithms. This differs completely from the method used by the author recently in the proof of the same kind of statement, where class field theory was applied.

Linear forms in logarithms have an important role in the theory of Diophantine equations. In this paper, we prove explicit [Formula: see text]-adic lower bounds for linear forms in [Formula: see text]-adic logarithms of rational numbers using Padé approximations of the second kind.

A linear form in logarithms of algebraic numbers is an expression of the form $$ \beta _1 \log \alpha _1 + \cdots + \beta _n log \alpha _n , $$ where the α’s and the β’s denote complex algebraic numbers, and log denotes any determination of the logarithm.

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

We apply inequalities from the theory of linear forms in logarithms to deduce effective results on [math] -integral points on certain higher-dimensional varieties when the cardinality of [math] is sufficiently small. These results may be viewed as a higher-dimensional version of an effective result of Bilu on integral points on curves. In particular, we prove a completely explicit result for integral points on certain affine subsets of the projective plane. As an application, we generalize an effective result of Vojta on the three-variable unit equation by giving an effective solution of the polynomial unit equation [math] , where [math] , [math] , and [math] are [math] -units, [math] , and [math] is a polynomial satisfying certain conditions (which are generically satisfied). Finally, we compare our results to a higher-dimensional version of Runge’s method, which has some characteristics in common with the results here.

For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α1, …, αn all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n0(D) which is explicitly computed, the inequalities (in the standard notation)have no solution in rational integers b1, …, bn (bn ≠ 0) of absolute value at most B, whenever n ≥ n0(D). The very favourable dependence on n is particularly useful.

We indicate a number of qualifications and amendments that are necessary so as to correct the statements and proofs of the theorems in our papers “Computing the effectively computable bound in Baker's inequality for linear forms in logarithms”, 15 (1976), 33–57, and its sequel “Multiplicative relations in number fields”, 16 (1977), 83–98, and remark on recent observations that would yield yet sharper results.

Alan Baker, who died on the 4th of February of this year, was born in England on the 19th of August 1939. In 1965 he defended his doctoral dissertation titled ‘Some Aspects of Diophantine Approximation’ at Trinity College Cambridge under the guidance of Harold Davenport. It is a very unusual fact that eight of his papers, including [1], which is discussed in the next section, had appeared in print before he submitted his doctoral dissertation. Baker was awarded the Fields Medal in 1970 at the International Congress of Mathematicians at Nice. Other honors he received are detailed in the Article-in-a-Box in this issue. In the present text, we will discuss his paper [1] and the ‘Baker theory of linear forms in logarithms’, which started with the series of four papers [2, 3, 4, 5] published in the British journal Mathematika. This new theory was an impressive breakthrough in the field of Diophantine approximation and we will briefly present some of its applications. For precise references to original papers, including those mentioned in the next sections, the reader is directed to the monographs [6, 7, 8, 9].