---
tag: XZBPKQWD0P
kind: concept
name: Toric ideal
msc: [13F65, 14M25]
sources:
  - url: https://en.wikipedia.org/wiki/Toric_ideal
same_as:
  wikidata: Q131758875
  wikipedia: Toric ideal
---

## Summary

A toric ideal is the ideal of all polynomial relations among a finite set of monomials in Laurent variables.
It is a prime ideal generated by binomials, that is, by differences of two monomials.
Its zero set is an affine toric variety, so toric ideals connect commutative algebra with polyhedral geometry and with integer programming and algebraic statistics.

## Explanations

### L1 human

Take some monomials, such as $s$, $st$ and $st^2$.
They satisfy relations: here $s \cdot st^2 = (st)^2$.
Naming the monomials $x_1, x_2, x_3$, the relation becomes $x_1 x_3 - x_2^2 = 0$.
The toric ideal collects all such relations, and each relation says that two products of the monomials have the same exponent vector.
So finding the toric ideal is a question about linear relations among integer vectors.