{{short description|Points of small height in projective space lie in a finite number of hyperplanes}} In mathematics, the '''subspace theorem''' says that points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained by {{harvs|txt|authorlink=Wolfgang M. Schmidt|first=Wolfgang M. |last=Schmidt|year= 1972}}.
==Statement== The subspace theorem states that if ''L''<sub>1</sub>,...,''L''<sub>''n''</sub> are linearly independent linear forms in ''n'' variables with algebraic coefficients and if ε>0 is any given real number, then the non-zero integer points ''x'' with :<math>|L_1(x)\cdots L_n(x)|<|x|^{-\epsilon}</math> lie in a finite number of proper subspaces of '''Q'''<sup>''n''</sup>.
A quantitative form of the theorem, which determines the number of subspaces containing all solutions, was also obtained by Schmidt, and the theorem was generalised by {{harvtxt|Schlickewei|1977}} to allow more general absolute values on number fields.
==Applications== The theorem may be used to obtain results on Diophantine equations such as Siegel's theorem on integral points and solution of the S-unit equation.<ref>Bombieri & Gubler (2006) pp. 176–230.</ref>
===A corollary on Diophantine approximation=== The following corollary to the subspace theorem is often itself referred to as the ''subspace theorem''. If ''a''<sub>1</sub>,...,''a''<sub>''n''</sub> are algebraic such that 1,''a''<sub>1</sub>,...,''a''<sub>''n''</sub> are linearly independent over '''Q''' and ε>0 is any given real number, then there are only finitely many rational ''n''-tuples (''x''<sub>1</sub>/y,...,''x''<sub>''n''</sub>/y) with :<math>|a_i-x_i/y|<y^{-(1+1/n+\epsilon)},\quad i=1,\ldots,n.</math>
The specialization ''n'' = 1 gives the Thue–Siegel–Roth theorem. One may also note that the exponent 1+1/''n''+ε is best possible by Dirichlet's theorem on diophantine approximation.
==References== {{Reflist}} * {{cite book | first1=Enrico | last1=Bombieri | authorlink1=Enrico Bombieri | first2=Walter | last2=Gubler | title=Heights in Diophantine Geometry | series=New Mathematical Monographs | volume=4 | publisher=Cambridge University Press | location=Cambridge | year=2006 | isbn=978-0-521-71229-3 | zbl=1130.11034 | mr=2216774}} * {{cite journal | last=Schlickewei | first=Hans Peter | authorlink=Hans Peter Schlickewei | title=On norm form equations | journal=J. Number Theory | doi=10.1016/0022-314X(77)90072-5 | year=1977 | volume=9 | issue=3 | pages=370–380 | mr=0444562 | doi-access=free }} * {{cite journal | last1=Schmidt | first1=Wolfgang M. | authorlink=Wolfgang M. Schmidt | title=Norm form equations | mr=0314761 | year=1972 | journal=Annals of Mathematics |series=Second Series | volume=96 | pages=526–551 | issue=3 | doi=10.2307/1970824 | jstor=1970824 }} * {{cite book | last=Schmidt | first=Wolfgang M. | authorlink=Wolfgang M. Schmidt | title=Diophantine Approximation | series=Lecture Notes in Mathematics | volume=785 | publisher=Springer-Verlag | year=1980 | edition=1996 with minor corrections | zbl=0421.10019 | mr=568710 | doi=10.1007/978-3-540-38645-2 | isbn=3-540-09762-7 | location=Berlin}} * {{cite book | last=Schmidt | first=Wolfgang M. | authorlink=Wolfgang M. Schmidt | title=Diophantine Approximations and Diophantine Equations | series=Lecture Notes in Mathematics | volume=1467 | publisher=Springer-Verlag | year=1991 | location=Berlin | isbn=3-540-54058-X | zbl=0754.11020 | mr=1176315 | doi=10.1007/BFb0098246| s2cid=118143570 }}
Category:Diophantine approximation Category:Theorems in number theory