grandes-ecoles 2023 Q16

grandes-ecoles · France · centrale-maths1__official Proof Proof That a Map Has a Specific Property

Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$. We recall that the degree of the zero polynomial is by convention equal to $-1$.
Show that there exists a natural number $n(T)$ such that, for every polynomial $p \in \mathbb{K}[X]$, $$\deg(Tp) = \max\{-1, \deg(p) - n(T)\}$$

Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$. We recall that the degree of the zero polynomial is by convention equal to $-1$.

Show that there exists a natural number $n(T)$ such that, for every polynomial $p \in \mathbb{K}[X]$,
$$\deg(Tp) = \max\{-1, \deg(p) - n(T)\}$$

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