grandes-ecoles 2012 QI.A

grandes-ecoles · France · centrale-maths2__mp Sequences and Series Recurrence Relations and Sequence Properties

Let $x$ be a linear recurrent sequence. Show that the set $J_x$ of polynomials $A$ such that $A(\sigma)(x) = 0$ is an ideal of $\mathbb{K}[X]$, not reduced to $\{0\}$.
We recall that this implies two things:

on the one hand, there exists in $J_x$ a unique monic polynomial $B$ of minimal degree;
on the other hand, the elements of $J_x$ are the multiples of $B$.

By definition, we say that $B$ is the minimal polynomial of the sequence $x$, that the degree of $B$ is the minimal order of $x$, and that the relation $B(\sigma)(x) = 0$ is the minimal recurrence relation of $x$.

Let $x$ be a linear recurrent sequence. Show that the set $J_x$ of polynomials $A$ such that $A(\sigma)(x) = 0$ is an ideal of $\mathbb{K}[X]$, not reduced to $\{0\}$.

We recall that this implies two things:
\begin{itemize}
  \item on the one hand, there exists in $J_x$ a unique monic polynomial $B$ of minimal degree;
  \item on the other hand, the elements of $J_x$ are the multiples of $B$.
\end{itemize}
By definition, we say that $B$ is the minimal polynomial of the sequence $x$, that the degree of $B$ is the minimal order of $x$, and that the relation $B(\sigma)(x) = 0$ is the minimal recurrence relation of $x$.

Paper Questions

QI.A QI.B.1 QI.B.2 QI.C.1 QI.C.2 QI.C.3 QI.D QII.A.1 QII.A.2 QII.B.1 QII.B.2 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.C.1 QIII.C.2 QIII.D.1 QIII.D.2 QIII.D.3