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If $f:A\subset\mathbb{R}^n\to\mathbb{R}$ is in $C^{n+1}(A)$ with $A$ open then fixed $x_0\in A$ $f(x_0+h) = \sum_{k=0}^n\frac{1}{k!}\sum(D_{i_1\dots i_k}f)(x_0)h_{i_1}\cdots h_{i_k} + \operatorname{o}(|h|^n)$ where the inner sum has $n^k$ terms, one for each partial derivative of order $k$. Playing with this inner sum I noticed that the $k$th term is a homogeneous polynomial of degree $k$ in $\mathbb{R}[h_1,\dots,h_n]$. Looking for a compact algebraic form for the generic term I observed that for:

  • $k=0$ it's $f(x_0)$, a nice number
  • $k=1$ it's $\sum(D_{i_1}f)(x_0)h_{i_1}$, the dot product of the vector $[(D_1f)(x_0),\dots, (D_nf)(x_0)]$ with $h$
  • $k=2$ it's $\sum(D_{i_1i_2}f)(x_0)h_{i_1}h_{i_2}$, the scalar product of $h$ with itself respect to the product defined by the matrix $[(D_jf_i)(x_0)]_{ij}$

Are there some algebraic objects/operations to accomplish this for $k=n$?

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Sure. A homogeneous polynomial function of degree $d$ on a vector space $V$ is just an element of the symmetric power $S^d(V^{\ast})$. You can represent these things, if you really want to, using symmetric tensors. For $d = 1$ this just gives an element of $V^{\ast}$, for $d = 2$ this gives a "matrix" (although it is somewhat dangerous IMO to think in these terms because one is making two choices), and so forth.