# homogeneous polynomial

Let $R$ be an associative ring. A (multivariate) polynomial $f$ over $R$ is said to be homogeneous of degree $r$ if it is expressible as an $R$-linear combination (http://planetmath.org/LinearCombination) of monomials of degree $r$:

 $f(x_{1},\ldots,x_{n})=\sum_{i=1}^{m}a_{i}{x_{1}}^{r_{i1}}\cdots{x_{n}}^{r_{in}},$

where $r=r_{i1}+\cdots+r_{in}$ for all $i\in\{1,\ldots,m\}$ and $a_{i}\in R$.

A general homogeneous polynomial is also known sometimes as a polynomial form. A homogeneous polynomial of degree 1 is called a linear form; a homogeneous polynomial of degree 2 is called a quadratic form; and a homogeneous polynomial of degree 3 is called a cubic form.

Remarks.

1. 1.

If $f$ is a homogeneous polynomial over a ring $R$ with $\operatorname{deg}(f)=r$, then $f(tx_{1},\ldots,tx_{n})=t^{r}f(x_{1},\ldots,x_{n})$. In fact, a homogeneous function that is also a polynomial is a homogeneous polynomial.

2. 2.

Every polynomial $f$ over $R$ can be expressed uniquely as a finite sum of homogeneous polynomials. The homogeneous polynomials that make up the polynomial $f$ are called the homogeneous components of $f$.

3. 3.

If $f$ and $g$ are homogeneous polynomials of degree $r$ and $s$ over a domain $R$, then $fg$ is homogeneous of degree $r+s$. From this, one sees that given a domain $R$, the ring $R[\boldsymbol{X}]$ is a graded ring, where $\boldsymbol{X}$ is a finite set of indeterminates. The condition that $R$ does not have any zero divisors is essential here. As a counterexample, in $\mathbb{Z}_{6}[x,y]$, if $f(x,y)=2x+4y$ and $g(x,y)=3x+3y$, then $f(x,y)g(x,y)=0$.

Examples

• $f(x,y)=x^{2}+xy+yx+y^{2}$ is a homogeneous polynomial of degree 2. Notice the middle two monomials could be combined into the monomial 2xy if the variables are allowed to commute with one another.

• $f(x)=x^{3}+1$ is not a homogeneous polynomial.

• $f(x,y,z)=x^{3}+xyz+zyz+3xy^{2}+x^{2}-xy+y^{2}+zy+z^{2}+xz+y+2x+6$ is a polynomial that is the sum of four homogeneous polynomials: $x^{3}+xyz+zyz+3xy^{2}$ (with degree 3), $x^{2}-xy+y^{2}+zy+z^{2}+xz$ (degree = 2), $y+2x$ (degree = 1) and $6$ (deg = 0).

• Every symmetric polynomial can be written as a sum of symmetric homogeneous polynomials.

 Title homogeneous polynomial Canonical name HomogeneousPolynomial1 Date of creation 2013-03-22 14:53:42 Last modified on 2013-03-22 14:53:42 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 17 Author CWoo (3771) Entry type Definition Classification msc 16R99 Classification msc 13B25 Classification msc 16S36 Classification msc 11E76 Synonym polynomial form Related topic HomogeneousIdeal Related topic HomogeneousFunction Related topic HomogeneousEquation Defines homogeneous component Defines cubic form Defines linear form