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homogeneous polynomial (Definition)

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$:

$\displaystyle f(x_1,\ldots,x_n)=\sum_{i=1}^{m}a_i{x_1}^{r_{i1}}\cdots{x_n}^{r_{in}},$
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.



"homogeneous polynomial" is owned by CWoo.
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See Also: homogeneous ideal, homogeneous function, homogeneous equation

Other names:  polynomial form
Also defines:  homogeneous component, cubic form, linear form

Attachments:
homogeneous equation (Topic) by pahio
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Cross-references: symmetric, symmetric polynomial, variables, counterexample, zero divisors, indeterminates, finite set, graded ring, domain, sum, finite, homogeneous function, quadratic form, monomials, expressible, homogeneous of degree, polynomial, ring, associative
There are 13 references to this entry.

This is version 14 of homogeneous polynomial, born on 2004-12-14, modified 2006-02-24.
Object id is 6577, canonical name is HomogeneousPolynomial.
Accessed 9133 times total.

Classification:
AMS MSC: 13B25 (Commutative rings and algebras :: Ring extensions and related topics :: Polynomials over commutative rings)
 16R99 (Associative rings and algebras :: Rings with polynomial identity :: Miscellaneous)
 16S36 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Ordinary and skew polynomial rings and semigroup rings)
 11E76 (Number theory :: Forms and linear algebraic groups :: Forms of degree higher than two)
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