Proof. [00LW]

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Proof.

Take a complete non-Archimedean valued field extension $(K,\lvert\mathord{\cdot}\rvert_{K})$ of $(k,\lvert\mathord{\cdot}\rvert_{k})$ such that

\forall i\in\{0,\dots,d\},\quad\{r_{i}\}_{i\in\{0,\dots,d\}}\subseteq\lvert k^{\times}\rvert_{K},

hence for any $i\in\{0,\dots,d\}$ , there exist elements $\kappa_{i}\in K$ such that $\lvert\kappa_{i}\rvert_{K}=r_{i}$ . One denotes by $x(\boldsymbol{r})\in(\mathbb{P}^{d}_{k})^{\mathrm{an}}$ the point given by coordinates $[\kappa_{0}:\dots:\kappa_{d}]$ .

Claim 5.2.

For any $i\in\{0,\dots,d\}$ , one has

\lVert T_{i}\rVert_{\phi}=r_{i}=\lvert T_{i}\rvert_{\phi}(x(\boldsymbol{r})).

In other words, the maximum of the function $\lvert T_{i}\rvert_{\phi}(x)$ on $(\mathbb{P}^{d}_{k})^{\mathrm{an}}$ is $r_{i}$ , and the maximum values of these $d+1$ functions can be attained at the same point $x(\boldsymbol{r})$ .

Proof.

By the orthogonality of the basis $\{T_{i}\}_{i\in\{0,\dots,d\}}$ , we can compute

\begin{split}\lvert T_{i}\rvert_{\phi}(x(\boldsymbol{r}))&=\inf_{\begin{subarray}{c}(\sum_{m\in\{0,\dots,d\}}f_{m}\cdot T_{m})(x(\boldsymbol{r}))=(T_{i})(x(\boldsymbol{r}))\\ (f_{0},\dots,f_{d})\in k^{d+1}\end{subarray}}\Big\lVert\sum_{m\in\{0,\dots,d\}}f_{m}\cdot T_{m}\Big\rVert_{\phi}\\ &=\inf\limits_{\sum_{m\in\{0,\dots,d\}}f_{m}\kappa_{m}=\kappa_{i}}\max_{m\in\{0,\dots,d\}}\Big\{\lVert f_{m}\cdot T_{m}\rVert_{\phi}\Big\}\\ &=\inf\limits_{\sum_{m\in\{0,\dots,d\}}f_{m}\kappa_{m}=\kappa_{i}}\max_{m\in\{0,\dots,d\}}\Big\{\lvert f_{m}\rvert\lvert\kappa_{m}\rvert\Big\}\\ &=\inf\limits_{\sum_{m\in\{0,\dots,d\}}f_{m}(\kappa_{m}/\kappa_{i})=1}\max_{m\in\{0,\dots,d\}}\Big\{\lvert f_{m}\rvert\lvert\kappa_{m}/\kappa_{i}\rvert\cdot\lvert\kappa_{i}\rvert\Big\}\\ &=\lvert\kappa_{i}\rvert=r_{i}.\end{split}

The last equality is obtained by Lemma 3.13. ∎

By this Claim, for any multi-index $J$ , the function $\lvert\boldsymbol{T}^{J}\rvert_{\phi}(x)$ can attain its maximum value $\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}$ at the point $x(\boldsymbol{r})\in(\mathbb{P}_{k}^{d})^{an}$ as the product of maximum of factors of the monomial. By definition,

\lVert\boldsymbol{T}^{J}\rVert_{|J|\phi}=\sup_{x\in(\mathbb{P}^{d})^{an}}\lvert\boldsymbol{T}^{J}\rvert_{\phi}(x)=\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}=\prod_{i\in\{0,\dots,d\}}\lVert T_{i}\rVert_{\phi}^{j_{i}}.

∎