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:

    “We can’t nourish our children if we don’t nourish ourselves.... Parents who manage to stay married, sane, and connected to each other share one basic characteristic: The ability to protect even small amounts of time together no matter what else is going on in their lives.”
    —Ron Taffel (20th century)

    “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)