Baker theorem on linear forms in logarithms

E637308

mathematical theorem result in transcendental number theory

The Baker theorem on linear forms in logarithms is a fundamental result in transcendental number theory that provides explicit lower bounds for nonzero linear combinations of logarithms of algebraic numbers, with powerful applications to Diophantine equations and Diophantine approximation.

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Observed surface forms (2)

Surface form	Occurrences
Baker’s theory on linear forms in logarithms	1
Linear Forms in the Logarithms of Algebraic Numbers	1

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in transcendental number theory ⓘ
appearsIn	Alan Baker’s work on transcendental number theory ⓘ monographs on Diophantine approximation ⓘ textbooks on transcendental number theory ⓘ
appliesTo	nonzero linear combinations of logarithms of algebraic numbers ⓘ
characterizedBy	effectivity of the bounds obtained ⓘ
concerns	explicit lower bounds for linear combinations of logarithms ⓘ linear forms in logarithms of algebraic numbers ⓘ
contributedTo	Alan Baker receiving the Fields Medal in 1970 ⓘ
field	Diophantine approximation ⓘ Diophantine equations NERFINISHED ⓘ number theory ⓘ transcendental number theory ⓘ
generalizes	earlier results of Gelfond and Schneider ⓘ
hasConsequence	bounds for exponents in exponential Diophantine equations can be made explicit ⓘ effective irrationality measures for certain algebraic numbers ⓘ effective lower bounds for linear forms in logarithms of algebraic numbers ⓘ many Diophantine equations have only finitely many integer solutions ⓘ
implies	linear forms in logarithms of algebraic numbers are rarely very small ⓘ
involves	algebraic number fields ⓘ degree of algebraic numbers ⓘ explicit constants depending on degrees and heights ⓘ heights of algebraic numbers ⓘ logarithms on the complex plane ⓘ
namedAfter	Alan Baker NERFINISHED ⓘ
provedBy	Alan Baker NERFINISHED ⓘ
provides	effective lower bounds for linear forms in logarithms ⓘ
relatedTo	Baker–Wüstholz theorem NERFINISHED ⓘ Gelfond–Schneider theorem NERFINISHED ⓘ Matveev’s theorem on linear forms in logarithms NERFINISHED ⓘ the theory of heights in Diophantine geometry ⓘ
status	fundamental tool in modern Diophantine analysis ⓘ
timePeriod	1960s ⓘ
usedFor	bounding integer solutions of S-unit equations ⓘ bounding integer solutions of Thue equations ⓘ bounding integer solutions of Thue–Mahler equations ⓘ bounding integer solutions of exponential Diophantine equations ⓘ effective finiteness results for Diophantine equations ⓘ effective results in Diophantine approximation ⓘ effective results on the Mordell equation ⓘ effective versions of Siegel’s theorem on integral points ⓘ results on Catalan-type equations ⓘ results on Pillai-type equations ⓘ results on perfect powers in recurrence sequences ⓘ results on the Lebesgue–Nagell equation ⓘ results on the Ramanujan–Nagell equation ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Diophantine approximation → hasKeyResult → Baker theorem on linear forms in logarithms ⓘ

Alan Baker → notableWork → Baker theorem on linear forms in logarithms ⓘ

this entity surface form: Baker’s theory on linear forms in logarithms

Alan Baker → notablePublication → Baker theorem on linear forms in logarithms ⓘ

this entity surface form: Linear Forms in the Logarithms of Algebraic Numbers