Homogeneous Polynomial - Basic Properties

Basic Properties

The number of different homogeneous monomials of degree M in N variables is

The Taylor series for a homogeneous polynomial P expanded at point x may be written as

$\begin{matrix} P(x+y)= \sum_{j=0}^n {n \choose j} P ( &\underbrace{x,x,\dots ,x}, & \underbrace{y,y,\dots ,y} ). \\ & j & n-j\\ \end{matrix}$

Another useful identity is

$\begin{matrix} P(x)-P(y)= \sum_{j=0}^{n-1} {n \choose j} P ( &\underbrace{y,y,\dots ,y}, & \underbrace{(x-y),(x-y),\dots ,(x-y)} ). \\ & j & n-j\\ \end{matrix}$

Read more about this topic: Homogeneous Polynomial

Famous quotes containing the words basic and/or properties:

“The basic idea which runs right through modern history and modern liberalism is that the public has got to be marginalized. The general public are viewed as no more than ignorant and meddlesome outsiders, a bewildered herd.”
—Noam Chomsky (b. 1928)

“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)