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, for $k, l \in \llbracket 0, n \rrbracket$, $$\delta^k\left(H_l\right)(0) = \begin{cases} 1 & \text{if } k = l \\ 0 & \text{if } k \neq l \end{cases}$$

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, for every $P \in \mathbb{R}_n[X]$, $$P = \sum_{k=0}^{n} \left(\delta^k(P)\right)(0) H_k$$

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Give the coordinates of the polynomial $X^3 + 2X^2 + 5X + 7$ in the basis $(H_0, H_1, H_2, H_3)$ of $\mathbb{R}_3[X]$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Deduce a polynomial $P \in \mathbb{R}_5[X]$ such that $$\delta^2(P) = X^3 + 2X^2 + 5X + 7$$

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Determine the real sequences $\left(u_k\right)_{k \in \mathbb{N}}$ such that $$u_{k+2} - 2u_{k+1} + u_k = k^3 + 2k^2 + 5k + 7 \quad (k \in \mathbb{N})$$

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 $k \in \mathbb{Z}$. Calculate $H_n(k)$. Distinguish three cases: $k \in \llbracket 0, n-1 \rrbracket$, $k \geqslant n$, and $k < 0$. For the latter case, set $k = -p$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Deduce that $H_n(\mathbb{Z}) \subset \mathbb{Z}$, that is, $H_n$ is integer-valued on the 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 $P \in \mathbb{R}_n[X]$ be integer-valued on the integers. Show that $\delta(P)$ is also integer-valued on the integers.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
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 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 $x , y$ be strictly positive vectors of $\mathbb { R } ^ { n }$ and let $S , R$ be two sign diagonal matrices.
(a) Show that $$( S x \mid R y ) \leq ( x \mid y )$$ with equality if and only if $R = S$.
(b) Prove the uniqueness of $S$ in Broyden's theorem.
(c) Show that $$\| S x + R y \| \leq \| x + y \|$$ with equality if and only if $R = S$.

Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $$J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$$ the supremum of $J_{f}$ on $\Sigma_{N}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$ the set of $p$ in $\Sigma_{N}$ for which the supremum is attained.
Show that $\Sigma_{N}(f)$ is non-empty.

Deduce from question 6 that $x > 0$ and $x + O x \geq 0$ as well as $x - O x \geq 0$. Conclude.

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.

Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Let $p \in \Sigma_{N}$.
(a) Suppose that $p_{1} = 0$ and $p_{2} > 0$. Show that there exists $p'$ in $\Sigma_{N}$ such that $J_{f}(p') > J_{f}(p)$ (you may look for $p'$ close to $p$).
(b) Deduce that if $p \in \Sigma_{N}(f)$, then $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$.

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 $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Let $p \in \Sigma_{N}$. We now assume that $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$. We denote $E_{0} = \{a \in \mathbb{R}^{N} \mid \sum_{i=1}^{N} a_{i} = 0\}$.
(a) Verify that $E_{0}$ is a vector subspace of $\mathbb{R}^{N}$ and give its dimension. Identify the orthogonal $E_{0}^{\perp}$ of $E_{0}$ for the canonical inner product on $\mathbb{R}^{N}$.
(b) Let $a \in E_{0}$ and $\tilde{p} : \mathbb{R} \rightarrow \mathbb{R}^{N}$ defined by $\tilde{p}(t) = p + ta$. Show that there exists $\epsilon > 0$ such that $\tilde{p}(t) \in \Sigma_{N}$ for all $t \in ]-\epsilon, \epsilon[$. Calculate the derivative of $\tilde{p}$ at 0.
(c) Suppose further that $p \in \Sigma_{N}(f)$. Show that for all $a \in E_{0}$, we have $\sum_{i=1}^{N} a_{i}(f_{i} - \ln(p_{i})) = 0$. Deduce that there exists $c \in \mathbb{R}$ such that $\ln(p_{i}) = f_{i} + c$ for all $i \in \{1, \ldots, N\}$.

Throughout this question, $\mathcal{S}$ is a simplex in $\mathbb{R}^n$ such that $0 \in \mathring{\mathcal{S}}$. We want to show that $$\operatorname{Card}\left(\mathring{\mathcal{S}} \cap \mathbb{Z}^n\right) \geqslant 2\left\lfloor \operatorname{Vol}(\mathcal{S})\left(\frac{a(\mathcal{S})}{a(\mathcal{S})+1}\right)^n \right\rfloor + 1$$ We then set $a = a(\mathcal{S})$, and $k = \rfloor \operatorname{Vol}(\mathcal{S})\left(\frac{a}{a+1}\right)^n \lfloor$.
10a. Express, for $\beta \in \mathbb{R}^*$ and $x \in \mathbb{R}^n$, $\operatorname{Vol}(\beta \mathcal{S})$ and $\operatorname{Vol}(\mathcal{S} - x)$. Show that for $\lambda \in [0,1[$ sufficiently close to $1$, $\operatorname{Vol}\left(\frac{\lambda a}{a+1}\mathcal{S}\right) > k$.
10b. For $\lambda$ as in the previous question, let $v_0, \ldots, v_k$ be the $k+1$ distinct points in $\frac{\lambda a}{a+1}\mathcal{S}$ satisfying $v_i - v_j \in \mathbb{Z}^n$ for all $i, j$, whose existence is guaranteed by Theorem 1. Show that the points $v_i - v_j$ are in $\lambda \mathcal{S}$. Deduce that the $v_i - v_j$ are in $\mathring{\mathcal{S}}$.
10c. Show that there exists an index $j \in \{0, \ldots, k\}$ such that the $(2k+1)$ points $0, \pm(v_i - v_j)$, for $i \in \{0, \ldots, k\} \setminus \{j\}$ are distinct. Deduce the statement of question 10, then that $$\operatorname{Card}\left(\mathring{\mathcal{S}} \cap \mathbb{Z}^n\right) \geqslant \operatorname{Vol}(\mathcal{S})\left(\frac{a(\mathcal{S})}{2}\right)^n$$

Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Identify $\Sigma_{N}(f)$. Show that $J_{f,*} = \ln\left(\sum_{i=1}^{N} e^{f_{i}}\right)$.

Deduce from Broyden's theorem that there exist a strictly positive vector $x$ and a sign diagonal matrix $S$ such that $O x = S x$ and deduce that $u = x + S x$ is the positive vector of Tucker's theorem.

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 $nx + k(y-x) \in \Lambda$.