---
tag: 2C3PAQ7H50
kind: concept
name: Homogeneous polynomial
msc: [13F20, "12E10", 13A02]
specializes: [07GNX8GCYP]
sources:
  - url: https://en.wikipedia.org/wiki/Homogeneous_polynomial
same_as:
  wikidata: Q1474074
  wikipedia: Homogeneous polynomial
---

## Summary

A homogeneous polynomial is a [[polynomial in several variables|07GNX8GCYP]] whose nonzero terms all have the same total degree.
Such polynomials are also called forms; the [[quadratic forms|0RBQS1JJCR]] are the homogeneous polynomials of degree $2$.
A homogeneous polynomial of degree $d$ satisfies $f(\lambda x) = \lambda^d f(x)$, so its zero set is a union of lines through the origin.
This is why homogeneous polynomials describe the subsets of projective space, and why they appear in the definition of [[quasi-algebraically closed fields|G7RM16N4PF]].

## Explanations

### L1 human

Take a polynomial and add the exponents in each term.
If every term gives the same total, the polynomial is homogeneous.
For example, $x^3 + 2xy^2 - 5y^3$ is homogeneous of degree $3$, since each term has total degree $3$.
The polynomial $x^2 + y$ is not homogeneous, because $x^2$ has degree $2$ and $y$ has degree $1$.
If you scale all variables by $2$, a homogeneous polynomial of degree $3$ is scaled by $2^3 = 8$.