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 $r$:
$$f({x}_{1},\mathrm{\dots},{x}_{n})=\sum _{i=1}^{m}{a}_{i}x_{1}{}^{{r}_{i1}}\mathrm{\cdots}x_{n}{}^{{r}_{in}},$$ 
where $r={r}_{i1}+\mathrm{\cdots}+{r}_{in}$ for all $i\in \{1,\mathrm{\dots},m\}$ 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.
If $f$ is a homogeneous polynomial over a ring $R$ with $\mathrm{deg}(f)=r$, then $f(t{x}_{1},\mathrm{\dots},t{x}_{n})={t}^{r}f({x}_{1},\mathrm{\dots},{x}_{n})$. In fact, a homogeneous function that is also a polynomial is a homogeneous polynomial.

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.
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 $R[\bm{X}]$ is a graded ring^{}, where $\bm{X}$ 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

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$f(x,y)={x}^{2}+xy+yx+{y}^{2}$ 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.

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

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

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Every symmetric polynomial^{} can be written as a sum of symmetric^{} homogeneous polynomials.
Title  homogeneous polynomial 
Canonical name  HomogeneousPolynomial1 
Date of creation  20130322 14:53:42 
Last modified on  20130322 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 