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

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