Gordan's lemma

Remarks

Let $P = \set{t_1 v_1 + \cdots + t_k v_k : 0 \le t_i < 1}$. It is a bounded set, so $P \cap \Z^n$ is finite. Every $x \in C \cap \Z^n$ can be written $x = \sum_i t_i v_i$ with $t_i \ge 0$, and then $x = \sum_i \floor{t_i} v_i + p$ with $p = \sum_i (t_i - \floor{t_i}) v_i \in P \cap \Z^n$. So the finite set $\set{v_1, \dots, v_k} \cup (P \cap \Z^n)$ generates $C \cap \Z^n$.

Example: $v_1 = (1, 0)$ and $v_2 = (1, 2)$ in $\Z^2$. Here $P \cap \Z^2 = \set{(0,0), (1,1)}$, so $C \cap \Z^2$ is generated by $(1,0)$, $(1,1)$, $(1,2)$. The point $(1,1)$ is needed: it lies in $C$ but is not a combination of $(1,0)$ and $(1,2)$ with nonnegative integer coefficients.

Rationality is needed. For the cone $C = \set{(x, y) : x \ge 0, \ 0 \le y \le \sqrt{2}\, x}$ the monoid $C \cap \Z^2$ is not finitely generated. Finitely many generators in $C \cap \Z^2$ only produce points of slope at most $q$, the largest slope of a generator, and $q < \sqrt{2}$ since $q$ is rational. But $C$ contains lattice points $(b, a)$ with $q < a/b < \sqrt{2}$.

Going deeper

For a student

The lemma is the combinatorial input of the theory of affine toric varieties. For a rational cone $\sigma$, the semigroup algebra of $\sigma^\vee \cap M$ is then finitely generated, and its spectrum is the toric variety of $\sigma$. The minimal generating set of $C \cap \Z^n$ for a pointed cone is its Hilbert basis. In invariant theory the same statement is used to show that rings of invariants of tori are finitely generated.