---
tag: HEKDP2TDNZ
kind: statement
role: lemma
name: Gordan's lemma
msc: [52B20, 14M25]
uses: [6VE4KA12VD, 0N6R902FG1, WN9V7WSJKF]
about: [6VE4KA12VD, 0N6R902FG1]
sources:
  - url: https://en.wikipedia.org/wiki/Gordan's_lemma
same_as:
  wikidata: Q25098650
  wikipedia: Gordan's lemma
---

## Statement

Let $n \ge 1$ and let $v_1, \dots, v_k \in \Z^n$.
Let $C = \set{\lambda_1 v_1 + \cdots + \lambda_k v_k : \lambda_1, \dots, \lambda_k \ge 0}$ be the rational polyhedral [[convex cone|6VE4KA12VD]] they generate in $\R^n$.
Then the [[monoid|0N6R902FG1]] $C \cap \Z^n$, with the addition of $\Z^n$, is finitely generated.

## 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 $q < a/b < \sqrt{2}$.

## Explanations

### L1 human

Take a cone in space whose edges point along whole-number directions.
The lattice points inside the cone can be added together, and the lemma says that finitely many of them generate all the others by addition.
The generators are the edge vectors together with the few lattice points in a small box spanned by them.

### L2 human

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|YWB981MEF2]] 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.