---
tag: 4G25S7PJ22
kind: concept
name: Monomial ideal
msc: [13F55, 13F20, "05E40"]
sources:
  - url: https://en.wikipedia.org/wiki/Monomial_ideal
same_as:
  wikidata: Q19369206
  wikipedia: Monomial ideal
---

## Summary

A monomial ideal is an ideal of a polynomial ring over a field that is generated by monomials.
Membership in a monomial ideal is a combinatorial question about exponent vectors, so many invariants of these ideals can be computed by counting.
Every ideal can be deformed to a monomial ideal, its ideal of leading terms with respect to a [[monomial order|E874NARWPV]].
This is the bridge from general ideals to combinatorics through [[Gröbner bases|C83KPT47R0]].

## Explanations

### L1 human

A monomial is a product of variables such as $x^2 y$.
A monomial ideal is what one gets by taking some monomials and all their multiples, then all sums of those.
Picture the exponents $(a, b)$ of the monomials $x^a y^b$ as points of a grid: a monomial ideal is a staircase region that is closed under moving up and to the right.
For example, $(x^2, xy, y^3)$ contains every monomial except $1, x, y, y^2$.
A polynomial lies in the ideal exactly when each of its terms lies in the staircase.