Subspace theorem

E637309

result in Diophantine approximation theorem in number theory

The Subspace theorem is a fundamental result in Diophantine approximation that describes how solutions to certain inequalities involving linear forms over algebraic numbers must lie in a finite union of proper subspaces.

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Statements (47)

Predicate	Object
instanceOf	result in Diophantine approximation ⓘ theorem in number theory ⓘ
alsoKnownAs	Schmidt Subspace theorem NERFINISHED ⓘ
appliesTo	Diophantine inequalities ⓘ S-unit equations ⓘ systems of linear forms ⓘ
assumption	coefficients lie in a number field ⓘ linear forms are linearly independent ⓘ solutions considered in integer or S-integer points ⓘ
centralConcept	proper linear subspaces of affine or projective space ⓘ
concerns	approximations to algebraic numbers ⓘ linear forms in several variables ⓘ
conclusion	outside a finite union of proper subspaces only finitely many solutions exist ⓘ solutions lie in finitely many proper subspaces ⓘ
describes	structure of solutions to certain Diophantine inequalities ⓘ
domain	algebraic number fields ⓘ
field	Diophantine approximation ⓘ number theory ⓘ
generalizes	Thue–Siegel–Roth theorem NERFINISHED ⓘ
hasVariant	Evertse–Schlickewei–Schmidt quantitative version ⓘ Schlickewei’s p-adic generalization NERFINISHED ⓘ absolute Subspace theorem NERFINISHED ⓘ p-adic Subspace theorem NERFINISHED ⓘ quantitative Subspace theorem NERFINISHED ⓘ
implies	Roth’s theorem on Diophantine approximation NERFINISHED ⓘ
inspired	applications to transcendence theory ⓘ developments in higher-dimensional Diophantine approximation ⓘ
involves	Archimedean and non-Archimedean valuations ⓘ inequalities with respect to a finite set of places ⓘ product of absolute values of linear forms ⓘ
namedAfter	Wolfgang M. Schmidt NERFINISHED ⓘ
originallyProvedBy	Wolfgang M. Schmidt NERFINISHED ⓘ
relatedTo	Diophantine geometry NERFINISHED ⓘ Vojta’s conjectures NERFINISHED ⓘ geometry of numbers ⓘ height functions on algebraic numbers ⓘ
strengthenedBy	Evertse–Schlickewei–Schmidt theorem NERFINISHED ⓘ Evertse’s quantitative refinements ⓘ Schlickewei’s quantitative refinements ⓘ
type	finiteness theorem ⓘ subspace-type Diophantine approximation theorem NERFINISHED ⓘ
typicalSetting	solutions in projective n-space over a number field ⓘ
usedFor	finiteness results for integral points on varieties ⓘ results on S-unit equations ⓘ results on exponential Diophantine equations ⓘ results on linear recurrence sequences ⓘ
yearProved	1972 ⓘ

Referenced by (1)

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

Diophantine approximation → hasKeyResult → Subspace theorem ⓘ