Siegel's lemma

Siegel's lemma guarantees a nonzero integer solution of bounded size to a system of homogeneous linear equations with integer coefficients. Its bounds depend on the number and size of the coefficients and the dimensions of the system.

Pigeonhole principle: A counting principle stating that placing more objects than containers forces at least one container to hold multiple objects. Siegel's lemma can be proved by mapping many bounded integer vectors to fewer possible equation outputs.
Homogeneous system of linear equations: A system of linear equations whose right-hand sides are all zero. Siegel's lemma concerns integer solutions to precisely these systems.
Carl Ludwig Siegel: A German mathematician whose work included major contributions to number theory and the theory of analytic functions. The lemma bears Siegel's name and arose within his work on Diophantine equations.
Transcendence theory: The branch of number theory studying numbers that are not roots of nonzero polynomials with integer coefficients. Small integer coefficients make auxiliary functions used in transcendence proofs possible.
Bombieri–Vaaler lemma: A refined form of Siegel's lemma giving bounds for a basis of integer solutions to linear equations. It strengthens the basic conclusion by controlling several independent solutions at once.
Geometry of numbers: The study of how geometric properties of sets constrain the lattice points they contain. Minkowski-style lattice arguments provide another route to bounded integer solutions.
Rank–nullity theorem: A theorem relating a linear map's domain dimension, kernel dimension, and image dimension. A sufficiently large variable space leaves a nontrivial rational kernel to search for integer points in.
Diophantine approximation: The study of approximating real or complex numbers by rational numbers or algebraic quantities. The lemma belongs to a tradition of deriving small integer relations through approximation.
Algebraic number theory: The study of number fields, their rings of integers, and related arithmetic structures. The lemma provides controlled relations among algebraic quantities and coefficients.
Siegel's lemma over number fields: A number-field version of Siegel's lemma that bounds the heights of nonzero solutions. It extends the integer-coefficient setting to equations over algebraic number fields.

Linked from 2 pages

Thue's lemmaCompared with: It extends the small-solution theme to exact linear equations rather than inequality constraints.