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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 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. 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. 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 $R[\boldsymbol{X}]$ 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.
Mathematics Subject Classification
16R99 no label found13B25 no label found16S36 no label found11E76 no label found Forums
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