Concept

Gordan's lemma

Summary
Gordan's lemma is a lemma in convex geometry and algebraic geometry. It can be stated in several ways.
  • Let A be a matrix of integers. Let M be the set of non-negative integer solutions of A \cdot x = 0. Then there exists a finite subset of vectors in M, such that every element of M is a linear combination of these vectors with non-negative integer coefficients.
  • The semigroup of integral points in a rational convex polyhedral cone is finitely generated.
  • An affine toric variety is an algebraic variety (this follows from the fact that the prime spectrum of the semigroup algebra of such a semigroup is, by definition, an affine toric variety).
The lemma is named after the mathematician Paul Gordan (1837–1912). Some authors have misspelled it as "Gordon's lemma". Proofs There are topological and algebraic proofs. Topological proof Let \sigma be the dual cone of the given rational po
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