Concours Centrale Supelec: mathématiques 1 MP

Fri May 26, 2023, par Pierre-Paul T. Views   250

A very long algebra problem set on polynomials and umbral calculus.

At the heart of this formalism is the Pincherle derivation.

We introduce the operator \(\mathbf{x}\) which is an endomorphism of polynomials: \[ \begin{alignat*}{1} \mathbf{x}:\mathbb{K}[X]&\to\mathbb{K}[X]\\ p&\mapsto Xp \end{alignat*} \]

We then introduce the Pincherle derivation, which is an endomorphism of polynomial endomorphisms: \[ \begin{alignat*}{1} \mathrm{End}(\mathbb{K}[X])&\to\mathrm{End}(\mathbb{K}[X])\\ T&\mapsto T'=T\mathbf{x}-\mathbf{x}T \end{alignat*} \]

We find familiar properties such as: \[(\sum_{k=0}^{+\infty} a_k D^k)'=\sum_{k=1}^{+\infty} ka_k D^{k-1}\] \[(ST)'=S'T+ST'\]

We are particularly interested in invariant shift endomorphisms: \[ \begin{alignat*}{1} E_a:\mathbb{K}[X]&\to\mathbb{K}[X]\\ p&\mapsto p(X+a) \end{alignat*} \] \[ T \text{ shift invariant }\Leftrightarrow \forall a,\quad TE_a=E_aT \]

Solutions to the Maths 1 test on May 9 in the MP Centrale Supelec 2023 exam:

Problem set:

For those who want to find out more, the following reference article is freely available (accessed May 26, 2023):