grandes-ecoles 2023 Q18

grandes-ecoles · France · centrale-maths1__official Proof Proof of Equivalence or Logical Relationship Between Conditions
Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$.
Show that the following three assertions are equivalent:
  1. [(1)] $T$ is invertible;
  2. [(2)] $T1 \neq 0$;
  3. [(3)] $\forall p \in \mathbb{K}[X], \deg(Tp) = \deg(p)$. 
Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$.

Show that the following three assertions are equivalent:
\begin{enumerate}
\item[(1)] $T$ is invertible;
\item[(2)] $T1 \neq 0$;
\item[(3)] $\forall p \in \mathbb{K}[X], \deg(Tp) = \deg(p)$.
\end{enumerate}
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