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 $i\in\lbrace 1,\ldots,m\rbrace$ 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 $f(tx_1,\ldots,tx_n)=t^rf(x_1,\ldots,x_n)$ . 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.