# Derivative of a homogeneous polynomial map

Let $K$ be a field and $V$ be a linear space over $K$. A map $p\colon V \to K$ is homogeneous polynomial of degree $n$ if there exist the symmetric $n$-linear form $f\colon V^{\times n}\to K$ such that $p(x)=f(x,x,\ldots,x)$ for any $x\in V$.

The question: where (what books or articles?) can I read about the map $\delta_a p\colon V \to K$, $a\in V$, such that $\delta_a p(x)=nf(a,x,\ldots,x)$ for any $x\in V$? As far as I understand, the map is the derivative of $p$ along $a$, is it? So I probably need some papers related to differential geometry?

Update 1: $\delta_a p(x)=f(a,x,\ldots,x)$ fixed, the correct one is $\delta_a p(x)=nf(a,x,\ldots,x)$ (sorry for the erratum)

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Have you tried to do some simple cases: two or three independent variables and degree two or three? –  Deane Yang Apr 24 '11 at 22:41
No, I have not. –  zimni pan Apr 24 '11 at 23:14
@Zimni Pan: you probably should then. Also, I think your question is more suitable for math.stackexchange.com than for here (see the FAQ for details). –  Willie Wong Apr 25 '11 at 0:24
For instance check this en.wikipedia.org/wiki/Homogeneous_function –  Pietro Majer Apr 25 '11 at 6:25
@Deane Yang: Thanks, I posted my question on math.stackexchange.com. Thanks! @Pietro Majer: I read the article, and found nothing about the map (derivative or whatever it is). But thanks for the link. –  zimni pan Apr 25 '11 at 7:23