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).

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