grandes-ecoles 2025 Q35

grandes-ecoles · France · x-ens-maths-d__mp Proof Existence Proof

Let $n \geq 1$ be an integer. Let $\mathbb{C}[[\mathbb{Z}^n]]$, $\mathcal{R}$, $\mathbb{C}(\mathbb{Z}^n)$, and $\mathrm{I} : \mathcal{R} \rightarrow \mathbb{C}(\mathbb{Z}^n)$ be as defined previously.
Let $u : \mathbb{Z}^n \rightarrow \mathbb{R}$ be an injective group homomorphism. Show that there exists a unique map $s_u : \mathbb{C}(\mathbb{Z}^n) \rightarrow \mathcal{R}$ satisfying the following three conditions:

[(a)] $s_u(Pf) = P\,s_u(f)$ for all $f \in \mathbb{C}(\mathbb{Z}^n)$ and $P \in \mathbb{C}[\mathbb{Z}^n]$.
[(b)] $\mathrm{I}(s_u(f)) = f$ for all $f \in \mathbb{C}(\mathbb{Z}^n)$.
[(c)] $s_u\left(\frac{1}{1-g}\right) = \sum_{n \in \mathbb{N}} g^n$ if $g$ is a finite linear combination of elements of the form $x^\gamma$ with $\gamma \in \mathbb{Z}^n$ satisfying $u(\gamma) > 0$.

Let $n \geq 1$ be an integer. Let $\mathbb{C}[[\mathbb{Z}^n]]$, $\mathcal{R}$, $\mathbb{C}(\mathbb{Z}^n)$, and $\mathrm{I} : \mathcal{R} \rightarrow \mathbb{C}(\mathbb{Z}^n)$ be as defined previously.

Let $u : \mathbb{Z}^n \rightarrow \mathbb{R}$ be an injective group homomorphism. Show that there exists a unique map $s_u : \mathbb{C}(\mathbb{Z}^n) \rightarrow \mathcal{R}$ satisfying the following three conditions:
\begin{itemize}
\item[(a)] $s_u(Pf) = P\,s_u(f)$ for all $f \in \mathbb{C}(\mathbb{Z}^n)$ and $P \in \mathbb{C}[\mathbb{Z}^n]$.
\item[(b)] $\mathrm{I}(s_u(f)) = f$ for all $f \in \mathbb{C}(\mathbb{Z}^n)$.
\item[(c)] $s_u\left(\frac{1}{1-g}\right) = \sum_{n \in \mathbb{N}} g^n$ if $g$ is a finite linear combination of elements of the form $x^\gamma$ with $\gamma \in \mathbb{Z}^n$ satisfying $u(\gamma) > 0$.
\end{itemize}

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24 Q25 Q26 Q27 Q28 Q29 Q30 Q31 Q32 Q33 Q34 Q35 Q36 Q37 Q38