This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| Baker–Wüstholz | |
|---|---|
| Name | Baker–Wüstholz |
| Field | Number theory |
| Known for | Effective lower bounds for linear forms in logarithms |
Baker–Wüstholz is a major result in transcendence theory and Diophantine approximation providing explicit effective lower bounds for nonzero linear forms in logarithms of algebraic numbers. The theorem strengthened earlier work by Alan Baker and refined techniques linking transcendence results used in solving exponential Diophantine equations, impacting work involving Thue–Mahler equations, Catalan’s conjecture, and effective finiteness results for S-unit equations. Influences range across interactions with the work of Gelfond, Schneider, Roth, and Siegel.
The context of Baker–Wüstholz lies in the study of linear forms λ = b_1 log α_1 + ... + b_n log α_n where each α_1, ..., α_n is an algebraic number and each b_1, ..., b_n is an integer or algebraic integer, with nonzero λ sought to be bounded away from zero effectively. Early milestones include Gelfond's and Schneider's results on transcendence, Baker's effective estimates, and contemporaneous progress by Mahler and Siegel. The statement gives an explicit lower bound |λ| > exp(-C(α_i,b_i, n) log B), where constants depend effectively on heights of α_i and degrees over Q, refining bounds by Baker and making them suitable for explicit Diophantine computations used by Tijdeman and Bugeaud. The theorem leverages heights introduced by Weil and refined by Northcott and Mahler.
The lineage starts from the Gelfond–Schneider theorem and the work of Thue and Siegel on Diophantine equations; later, Baker produced effective bounds in the 1960s that transformed certain existence proofs into explicit algorithms, influencing researchers like Waldschmidt, Lang, Silverman, Bugeaud, and Evertse. Wüstholz combined techniques from transcendence theory, algebraic geometry, and group varieties, building on the Subspace Theorem of Schmidt and methods related to Masser's work on linear forms in logarithms on commutative algebraic groups. Subsequent refinements involved contributions by Matveev, Yu Guangzhong, Hirata-Kohno, and Philippon; computational applications were pursued by Smart, Cohen, and Tzanakis. The interplay with modular methods and Galois representations connects historically to Ribet and later to efforts by Wiles and Taylor, though in distinct problems.
Proofs use interpolation determinants, auxiliary function constructions, and zero estimates on algebraic groups, combining ideas of Baker, Masser, and Wüstholz with height theory as developed by Néron and Tate. One builds an auxiliary polynomial with small values at logarithmic points using estimates akin to Siegel's lemma and applies an explicit form of the Schneider–Lang criterion and multiplicity estimates by Philippon and Nesterenko. Tools include the theory of commutative algebraic groups like G_m and elliptic curves studied by Silverman and Lang, as well as effective results for linear forms by Matveev and Baker–Wüstholz refinements. The argument further exploits Galois conjugation and height inequalities from Faltings and Northcott to control denominators and degrees, with crucial use of transcendence measures and explicit determinant bounds inspired by Hadamard and Hermite.
Effective lower bounds from Baker–Wüstholz enable explicit solution bounds for exponential Diophantine equations such as the Mordell equation, Thue equations, and S-unit equations studied by Evertse and Győry. They are instrumental in proving finiteness results used by Baker and Tijdeman for equations like Catalan-type problems resolved by Mihăilescu (Catalan's theorem) and in bounding integral points on curves treated by Siegel and Faltings. Computational number theory implementations by Cohen, Smart, and Bugeaud rely on these bounds in algorithms for determining unit groups in number fields and solving norm-form equations related to Dedekind and Dirichlet theory. Consequences extend to explicit forms of the Subspace Theorem applications seen in work by Evertse, Schlickewei, and Van der Poorten.
Classical examples include bounds for linear forms in two logarithms applied to the Pell equation and specific exponential Diophantine equations like Ramanujan–Nagell-type equations addressed by Le and Lehmer. Special cases encompass results for logarithms of conjugate algebraic units studied by Dirichlet and Kronecker, and effective results for linear forms on elliptic logarithms due to Baker and Coates and extended by Silverman and Langlands techniques. Matveev’s theorem provides comparable explicit constants in lower-dimension instances used by Bugeaud and Shorey for practical resolution of equations like x^2 + D = y^n studied by Lebesgue and Mignotte.
Generalizations include Matveev’s explicit bounds, Yu’s p-adic analogues, and extensions to linear forms on commutative algebraic groups by Wüstholz and collaborators. The Subspace Theorem of Schmidt and generalizations by Evertse and Schlickewei provide complementary finiteness frameworks; p-adic and adelic variants were developed by Baker', Yu, and Feldman. Connections to arithmetic geometry involve results by Faltings and Mordell–Weil theorem applications, while transcendence measures by Waldschmidt and Nesterenko extend the analytic toolkit. Current research threads link Baker–Wüstholz-type bounds with explicit methods in computational algebraic number theory and advances in transcendence due to Bilu, Habegger, and Pila.
Category:Transcendence theory