Tangent space to subspace of orbit in jet spaces

Question

I consider a map germ $f: (\mathbb{R}^n,0) \to \mathbb{R} $ which is $k$-determined for some $k \in \mathbb N$, i.e. for all map germs $g: (\mathbb{R}^n,0) \to \mathbb{R} $ having the same $k$-jet as $f$ , there exists a (local) diffeomorphism $\varphi: (\mathbb{R}^n,0) \to (\mathbb{R}^n,0)$ such that $g = f \circ \varphi$. Now consider the set \begin{align*} L := \{j^{k+1}(f+h) ~|~ h \in m^{k+1} \} \subseteq J^{k+1}_0(\mathbb R^n, \mathbb R),\end{align*} where $m^{k+1} = \{h : (\mathbb{R}^n,0) \to \mathbb{R}~|~ \frac{\partial^{\alpha}}{\partial x^{\alpha}}h(0) = 0, \forall \alpha \leq k \} $ and $J^{k+1}_0(\mathbb R^n, \mathbb R)$ is the $k$-jet space with source $0 \in \mathbb R^n$.

In the book "Singularities of Mappings" by David Mond and Juan J. Nuno-Ballesteros on page 48 they claim that for $z = j^{k+1}f(0)$ the tangent space of $L$ is \begin{align*} T_zL = m^{k+1}/m^{k+2}. \end{align*}

I don't see how this equality is obtained. How is the tangent space of $L$ at $z$ the space of monomials of degree $k+1$? Can anybody help me?

Answer 1

Those are not monomials. The space $m^{k+1}/m^{k+2}$ is the space of functions vanishing up to order $k$, modulo those vanishing up to order $k+1$, so Taylor series of homogeneous polynomials of degree exactly $k+1$. The space $L$ is, as you defined it, the translate of $m^{k+1}/m^{k+2}$ by $j^{k+1}f(0)$ inside $J^{k+1}_0$, i.e. $k+1$-jet of $f$ up to translation by $k+1$-jet of $h$.

The letter $m$ stands for maximal: the functions vanishing at the origin form a maximal ideal.