On Siegel's Lemma
Siegel's lemma in its simplest form is a statement about the existence of small-size solutions to a system of linear equations with integer coefficients: such results were originally motivated by their applications in transcendence. A modern version of this classical theorem guarantees the existence of a whole basis of small "size" for a vector space over a global field (that is number field, function field, or their algebraic closures). The role of size is played by a height function, an important tool from Diophantine geometry, which measures "arithmetic complexity" of points. For many applications it is also important to have a version of Siegel's lemma with some additional algebraic conditions placed on points in question. I will discuss the classical versions of Siegel's lemma, along with my recent results on existence of points of bounded height in a vector space outside of a finite union of varieties over a global field.
© 2008 Lenny Fukshansky
Fukshansky, Lenny. "On Siegel's Lemma." Algebra/Number Theory/Combinatorics Seminar, Claremont Colleges, Claremont, California. 23 September 2008.