subspace theorem
{{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 L1,...,Ln 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
:
lie in a finite number of proper subspaces of Qn.
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.Bombieri & Gubler (2006) pp. 176–230.
=A corollary on Diophantine approximation=
The following corollary to the subspace theorem is often itself referred to as the subspace theorem.
If a1,...,an are algebraic such that 1,a1,...,an are linearly independent over Q and ε>0 is any given real number, then there are only finitely many rational n-tuples (x1/y,...,xn/y) with
:
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 }}