homogeneous polynomialHomogeneousPolynomial2004-12-14 01:33:422006-02-24 18:42:22Definitionhomogeneous componentcubic formlinear form% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb,amscd}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here\PMlinkescapeword{homogeneous}
\PMlinkescapeword{degree}
Let $R$ be an associative ring. A (multivariate) polynomial $f$ over $R$ is said to be \emph{homogeneous of degree} $r$ if it is expressible as an $R$-\PMlinkname{linear combination}{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 $i\in\lbrace 1,\ldots,m\rbrace$ and $a_i\in R$.
A general homogeneous polynomial is also known sometimes as a \emph{polynomial form}. A homogeneous polynomial of degree 1 is called a \emph{linear form}; a homogeneous polynomial of degree 2 is called a \emph{quadratic form}; and a homogeneous polynomial of degree 3 is called a \emph{cubic form}.
\textbf{Remarks}.
\begin{enumerate}
\item 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.
\item 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 \emph{homogeneous components} of $f$.
\item 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 $f(x,y)g(x,y)=0$.
\end{enumerate}
\textbf{Examples}
\begin{itemize}
\item $f(x,y) = x^2+xy+yx+y^2$ 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.
\item $f(x) = x^3+1$ is not a homogeneous polynomial.
\item $f(x,y,z) = x^3+xyz+zyz+3xy^2+x^2-xy+y^2+zy+z^2+xz+y+2x+6$ 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).
\item Every symmetric polynomial can be written as a sum of symmetric homogeneous polynomials.
\end{itemize}