Gordan's lemma
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Gordan's lemma is a lemma in convex geometry and algebraic geometry. It can be stated in several ways.
- Let be a matrix of integers. Let be the set of non-negative integer solutions of . Then there exists a finite subset of vectors in , such that every element of is a linear combination of these vectors with non-negative integer coefficients.[1]
- The semigroup of integral points in a rational convex polyhedral cone is finitely generated.[2]
- 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 (1837ā1912). Some authors have misspelled it as "Gordon's lemma".