$(C,{\alpha})$-good on $V$, it is $(C',\alpha')$-good on $V'$ for every $C' \geq \max\{C,1\}$, $\alpha' \leq \alpha$ and $V'\subset V$.

One can note that from (G2), it follows that the supremum norm of a vector valued function ${\mathbf{f}}$ is $(C,{\alpha})$-good whenever each of its components is $(C,{\alpha})$-good. Furthermore, in view of (G3), we can replace the norm by an equivalent one, only affecting $C$ but not ${\alpha}$.

Polynomials in $d$ variables of degree at most $k$ defined on local fields can be seen to be $(C, 1/dk)$-good, with $C$ depending only on $d$ and $k$ using Lagrange interpolation. In [@KM], [@BKM] and [@KT] (for ultrametric fields), this property was extended to smooth functions satisfying certain properties. We rapidly recall, following [@S] (see also [@KT]), the definition of smooth functions in the ultrametric case. Let $U$ be a non-empty subset of $X$ without isolated points. For $n \in \mathbb{N}$, define $$\nabla^{n}(U) = \{(x_1,\dots,x_n) \in U, x_i \neq x_j \text{ for } i \neq j   \}.$$ The $n$-th order difference quotient of a function $f : U \to X$ is the function $\Phi_n(f) $ defined inductively by $\Phi_0 (f) =
f$ and, for $n \in {\mathbb{N}}$, and $(x_1,\dots,x_{n+1}) \in \nabla^n(U)$ by $$\Phi_{n}f(x_1,\dots,x_{n+1}) = 