Homogeneous Polynomials - 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 Polynomials

Famous quotes containing the words basic and/or properties:

    “Just as the constant increase of entropy is the basic law of the universe, so it is the basic law of life to be ever more highly structured and to struggle against entropy.”
    —Václav Havel (b. 1936)

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)