We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that $P \in \mathbb{R}_n[X]$ is integer-valued on the integers if and only if its coordinates in the basis $\left(H_k\right)_{k \in \llbracket 0, n \rrbracket}$ are integers.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Let $P \in \mathbb{R}[X]$ of degree $d \in \mathbb{N}$. Show that if $P$ is integer-valued on the integers then $d! P$ is a polynomial with integer coefficients. Study the converse.

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
Show that for every integer $k$ strictly positive, $k^\alpha$ belongs to $\mathbb{N}^*$.

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
Show that $\alpha$ is positive or zero.

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
We consider the application $f_\alpha$ defined on $\mathbb{R}_+^*$ by $f_\alpha(x) = x^\alpha$. Show that $\alpha$ is a natural number if and only if one of the successive derivatives of $f_\alpha$ vanishes at least at one strictly positive real.

We consider a real $\alpha$ such that for every prime number $p$, $p^\alpha$ is a natural number. We apply relation $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ to the function $f_\alpha(x) = x^\alpha$ and to the integer $n = \lfloor \alpha \rfloor + 1$ (where $\lfloor \cdot \rfloor$ denotes the floor function). We now choose $x \in \mathbb{N}^*$.
Show that the expression $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f_\alpha(x+j)$$ is a relative integer.

We consider a real $\alpha$ such that for every prime number $p$, $p^\alpha$ is a natural number. We apply relation $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ to the function $f_\alpha(x) = x^\alpha$ and to the integer $n = \lfloor \alpha \rfloor + 1$.
Conclude that $\alpha$ is a natural number.

Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The function $f$ is the pointwise limit of the sequence $f_n$. Show that the function $f$ satisfies the following equality on $\mathbb{R}$ $$f(x) = g(x) + \sum_{i=0}^{+\infty} p_i f\left(x - x_i\right) \tag{E}$$

Let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a bounded function that satisfies $h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$ for all $x \in \mathbb{R}$. Show that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, we have $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$.

Let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a bounded function that satisfies $h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$ for all $x \in \mathbb{R}$, and such that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$. Deduce that if moreover the support of $h$ is included in $\mathbb{R}^+$, then for all $x \in \mathbb{R}$, $h(x) = 0$.

Conclude that there exists a unique bounded function with support in $\mathbb{R}^+$ solution of $$f(x) = g(x) + \sum_{i=0}^{+\infty} p_i f\left(x - x_i\right) \tag{E}$$

Let $V \geqslant 0$ be a real number.
8a. Give an example of an integer simplex in $\mathbb{R}^2$ with volume greater than or equal to $V$ and having no interior integer points.
8b. Give an example of an integer simplex in $\mathbb{R}^3$ with volume greater than or equal to $V$ whose only integer points are the vertices.

Show that the set $\Lambda_X := \bigcup_{n \in \mathbb{N}} \left\{y \in \mathbb{R} \mid \mathbb{P}\left(S_n = y\right) > 0\right\}$ is countable and included in $\mathbb{R}^+$.

We are given an enumeration of the set $\Lambda_X = \{y_i \mid i \in \mathbb{N}\}$. Show that for all $x \in \mathbb{R}$, $$f_n(x) = \sum_{k=0}^{n} \sum_{i=0}^{+\infty} \mathbb{P}\left(S_k = y_i\right) g\left(x - y_i\right)$$

Let $\mathcal{K}$ be a compact convex set in $\mathbb{R}^n$ such that $0 \in \mathring{\mathcal{K}}$.
9a. Show that the set of $\lambda \geqslant 0$ such that $-\lambda \mathcal{K} \subset \mathcal{K}$ is an interval.
We denote $$a(\mathcal{K}) = \sup\{\lambda \geqslant 0 \mid -\lambda \mathcal{K} \subset \mathcal{K}\}$$
9b. Show that $a(\mathcal{K}) < \infty$ and that $a(\mathcal{K}) = \max\{\lambda \geqslant 0 \mid -\lambda \mathcal{K} \subset \mathcal{K}\}$.
9c. Show that $0 < a(\mathcal{K}) \leqslant 1$. Deduce that $a(\mathcal{K}) = 1$ if and only if $\mathcal{K}$ is symmetric with respect to 0.

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ such that $$\forall (x, y) \in \Lambda^2, \quad x + y \in \Lambda$$ We say that $\Lambda$ is closed under addition. Show that if $(x, y) \in \Lambda^2$, $(k, n) \in \mathbb{N} \times \mathbb{N}^*$ and $k \leqslant n$, then $nx + k(y-x) \in \Lambda$.

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Show that two integer simplexes $\mathcal{S}$ and $\mathcal{S}'$ are equivalent if and only if there exist a matrix $A \in \mathrm{GL}_n(\mathbb{Z})$ and a vector $b \in \mathbb{Z}^n$ such that $\mathcal{S}' = A(\mathcal{S}) - b$.

We prove Broyden's theorem by induction on the dimension. We assume the result holds up to rank $n - 1$ and we write $O$ in the form of a block matrix $$O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$$ where $P \in M _ { n - 1 } ( \mathbb { R } )$ and thus $r , q \in \mathbb { R } ^ { n - 1 }$ and $\alpha \in \mathbb { R }$.
Show that $| \alpha | \leq 1$ with equality if and only if $q = r = 0$.

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Show that the volume, the number of integer points and the number of interior integer points are the same for two equivalent integer simplexes.

We prove Broyden's theorem by induction on the dimension. We assume the result holds up to rank $n - 1$ and we write $O$ in the form of a block matrix $$O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$$ where $P \in M _ { n - 1 } ( \mathbb { R } )$ and thus $r , q \in \mathbb { R } ^ { n - 1 }$ and $\alpha \in \mathbb { R }$.
Treat the case $| \alpha | = 1$.

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) = 0$. Let $\eta > 0$. Show that there exists $A \geqslant 0$ such that for all $x > A$, $$\Lambda \cap [x, x + \eta] \neq \varnothing$$

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) = 0$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a uniformly continuous function. Suppose that for every sequence $\left(x_n\right)_{n \geqslant 0}$ with values in $\Lambda$ such that $x_n \rightarrow +\infty$, $f\left(x_n\right) \rightarrow 0$ when $n \rightarrow +\infty$. Show that $f(x) \rightarrow 0$ when $x \rightarrow +\infty$.

We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ where $O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$ with $P \in M _ { n - 1 } ( \mathbb { R } )$, $r , q \in \mathbb { R } ^ { n - 1 }$, $\alpha \in \mathbb { R }$.
Show that ${ } ^ { t } P P + q { } ^ { t } q = I _ { n - 1 }$, ${ } ^ { t } P r + \alpha q = 0$ and ${ } ^ { t } r r + \alpha ^ { 2 } = 1$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. We consider a function $h$ uniformly continuous and bounded on $\mathbb{R}$ such that for all $x \in \mathbb{R}$, $h(x) \leqslant h(0)$ and $$h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$$ We recall that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$. Show that for all $n \in \mathbb{N}$ and $x \geqslant 0$ such that $\mathbb{P}\left(S_n = x\right) > 0$, we have $h(-x) = h(0)$.

We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ where $O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$ with $P \in M _ { n - 1 } ( \mathbb { R } )$, $r , q \in \mathbb { R } ^ { n - 1 }$, $\alpha \in \mathbb { R }$.
Show that $${ } ^ { t } Q _ { + } Q _ { - } = I _ { n - 1 } - \frac { 2 } { 1 - \alpha ^ { 2 } } q { } ^ { t } q$$ and deduce that $$Q _ { - } = Q _ { + } - \frac { 2 } { 1 - \alpha ^ { 2 } } Q _ { + } q { } ^ { t } q$$