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
(http://planetmath.org/LinearCombination) of monomials
of degree r:
f(x1,…,xn)=m∑i=1aix1ri1⋯xnrin, |
where r=ri1+⋯+rin for all i∈{1,…,m} and ai∈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.
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1.
If f is a homogeneous polynomial over a ring R with deg(f)=r, then f(tx1,…,txn)=trf(x1,…,xn). 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[𝑿] is a graded ring
, where 𝑿 is a finite set of indeterminates. The condition that R does not have any zero divisors
is essential here. As a counterexample, in ℤ6[x,y], if f(x,y)=2x+4y and g(x,y)=3x+3y, then f(x,y)g(x,y)=0.
Examples
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•
f(x,y)=x2+xy+yx+y2 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.
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•
f(x)=x3+1 is not a homogeneous polynomial.
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•
f(x,y,z)=x3+xyz+zyz+3xy2+x2-xy+y2+zy+z2+xz+y+2x+6 is a polynomial that is the sum of four homogeneous polynomials: x3+xyz+zyz+3xy2 (with degree 3), x2-xy+y2+zy+z2+xz (degree = 2), y+2x (degree = 1) and 6 (deg = 0).
-
•
Every symmetric polynomial
can be written as a sum of symmetric
homogeneous polynomials.
Title | homogeneous polynomial |
Canonical name | HomogeneousPolynomial1 |
Date of creation | 2013-03-22 14:53:42 |
Last modified on | 2013-03-22 14:53:42 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 17 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 16R99 |
Classification | msc 13B25 |
Classification | msc 16S36 |
Classification | msc 11E76 |
Synonym | polynomial form |
Related topic | HomogeneousIdeal |
Related topic | HomogeneousFunction |
Related topic | HomogeneousEquation |
Defines | homogeneous component |
Defines | cubic form |
Defines | linear form |