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 Wolfgang M. Schmidt (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
\( {\displaystyle |L_{1}(x)\cdots L_{n}(x)|<|x|^{-\epsilon }} \)
lie in a finite number of proper subspaces of Qn.
A quantitative form of the theorem, in which the number of subspaces containing all solutions, was also obtained by Schmidt, and the theorem was generalised by 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.[1]
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
\( {\displaystyle |a_{i}-x_{i}/y|<y^{-(1+1/n+\epsilon )},\quad i=1,\ldots ,n.} \)
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
Bombieri & Gubler (2006) pp. 176–230.
Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. 4. Cambridge: Cambridge University Press. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. MR 2216774. Zbl 1130.11034.
Schlickewei, Hans Peter (1977). "On norm form equations". J. Number Theory. 9 (3): 370–380. doi:10.1016/0022-314X(77)90072-5. MR 0444562.
Schmidt, Wolfgang M. (1972). "Norm form equations". Annals of Mathematics. Second Series. 96 (3): 526–551. doi:10.2307/1970824. MR 0314761.
Schmidt, Wolfgang M. (1980). Diophantine approximation. Lecture Notes in Mathematics. 785 (1996 with minor corrections ed.). Berlin: Springer-Verlag. doi:10.1007/978-3-540-38645-2. ISBN 3-540-09762-7. MR 0568710. Zbl 0421.10019.
Schmidt, Wolfgang M. (1991). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. 1467. Berlin: Springer-Verlag. doi:10.1007/BFb0098246. ISBN 3-540-54058-X. MR 1176315. Zbl 0754.11020.
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