Exercise 33, § 1

For every integer $r\geqslant 1$, let $V_r=$ det(($x^j_i$)$_{1\leqslant i\leqslant r}$)$\in \mathbf{Z}[x_1, . . . , x_r]$ denote the Vandermonde determinant (A, III, p. 99, example $1$)$.^{0\leqslant j<r}$

a) Let $f_1, . . . ,f_r$ be functions indefinitely differentiable on $\mathbf{R}$. For all real numbers $x_1<\cdots < x_r$, there exist real numbers $(t_0, . . . , t_{r-1})$ in $[x_1, x_r]$ such that

det($f_i(x_j)$) $=$ 1!2! $\cdots 1(r-$ 1)! $V_r(x_1, . . . , x_r)$ det(($f_j^{(i)}(t_i)$)$_{01\leqslant\leqslant ij<r\leqslant r}$)

(cf. FVR, I, p. 48, Exercise 11).

b) Let $r$ be a positive integer and $m_1, . . . , m_r$ positive integers. Let

$\Delta_r=$ det(($x_i^{m_j}$)$_{1\leqslant i,j\leqslant r}$)$\in \mathbf{Z}[x_1, . . . , x_r]$.

The polynomial $\Delta_r$ is divisible by $V_r$ in $\mathbf{Z}[x_1, . . . , x_r]$. Let $F_r= \Delta_r/V_r$. c) Let $p$ be a prime number. Suppose that

Let $\xi_1,. . .,\xi_r$ be distinct $p$-th roots of unity. One has

(“Chebotarev theorem”; let $\xi$ be a primitive $p$-th root of unity; one has $F_r(\xi_1, . . . , \xi_r) = P(\xi )$ where $P\in \mathbf{Z}[x]$ is of degree $\leqslant p-1$; from the irreducibility of cyclotomic polynomials, the condition $F_r(\xi_1, . . . , \xi_r) = 0$ would imply that $F_r(1, . . . ,1)$ is divisible by $p$; using a) with $x_i\rightarrow 1$, prove that this is not the case.)

d) Let $G =\mathbf{Z}/p\mathbf{Z}$. For every function $f\not= 0$ from G into $\mathbf{C}$, one has

Card(Supp($f$)) $+$ Card(Supp($\widehat{f}$))$\geqslant p+ 1$.

e) For all non-empty subsets A and B in G such that

Card(A) + Card(B) $\geqslant p+ 1$,

there exists $f: G\rightarrow \mathbf{C}$ whose support is A and such that the support of $\widehat{f}$ is B. (Consider first the case where Card(A) + Card(B) $=p+ 1$.)

f) Let A and B be arbitrary subsets of G. One has

Card(A $+ B$)$\geqslant$ inf(Card(A) + Card(B) $-1, p$)

(“Cauchy-Davenport theorem”; apply the previous question to $(A,C)$ and to $(B,D)$, where C and D are subsets of cardinal respectively $p+ 1-$ Card(A) and $p+ 1-$ Card(B), such that $C\cap D$ has cardinal sup(Card(A) + Card(B) $-p,1$).)

g) Let $(a_1, . . . , a_{2p-1})$ be a family of elements of $\mathbf{F}_p$. There exists a subset $I\subset \{1, . . . ,2p-1\}$ of cardinal $p$ such that

This property is false in general for a family in $\mathbf{F}^{2p-2}_p$. (Let $0\leqslant b_1\leqslant \cdots \leqslant \cdots \leqslant b_{2p-1}< p$ be integers whose reductions modulo $p$ form a permutation of $(a_1, . . . , a_{2p-1})$; treat directly the case where there exists $i$ such that $a_i=a_{i+p-1}$, and apply the previous question in the contrary case.)

h) Let $k\geqslant 1$ be an integer. Let $\nu \geqslant 0$ be the integer such that $k=p^{\nu}r$ where $p$ does not divide $r$. Let us denote by $\gamma$ the integer

$\{\nu + 1$ if $p\geqslant 3$

$\nu + 2$ otherwise. If $\ell \geqslant 4k$ is an integer, then there exist integers $(n_1, . . . , n_{\ell})$ such that $p$ does not divide all the $n_i$ and such that

$n^k_1+\cdots +n^k_{\ell}\equiv 0$ mod$. p^{\gamma}$.