Let be an associative ring. A (multivariate) polynomial over is said to be homogeneous of degree if it is expressible as an -linear combination (http://planetmath.org/LinearCombination) of monomials of degree :
where for all and .
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.
If is a homogeneous polynomial over a ring with , then . In fact, a homogeneous function that is also a polynomial is a homogeneous polynomial.
Every polynomial over can be expressed uniquely as a finite sum of homogeneous polynomials. The homogeneous polynomials that make up the polynomial are called the homogeneous components of .
If and are homogeneous polynomials of degree and over a domain , then is homogeneous of degree . From this, one sees that given a domain , the ring is a graded ring, where is a finite set of indeterminates. The condition that does not have any zero divisors is essential here. As a counterexample, in , if and , then .
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.
is not a homogeneous polynomial.
is a polynomial that is the sum of four homogeneous polynomials: (with degree 3), (degree = 2), (degree = 1) and (deg = 0).
|Date of creation||2013-03-22 14:53:42|
|Last modified on||2013-03-22 14:53:42|
|Last modified by||CWoo (3771)|