On désigne par $r(n,m)$ le nombre de partitions de l’entier $n$ en parts supérieures ou égales à $m$, et $R(n,m)=r(n-m,m)$ le nombre de partitions de $n$ de plus petite part $m$. Dans un précédent article (voir [9]) un développement asymptotique de $r(n,m)$ est obtenu uniformément pour $1\le m=O\left(\sqrt{n}\right)$ ; on complète ce développement uniformément pour $1\le m=\left(n{log}^{-3}n\right)$. Afin de prolonger les résultats jusqu’à $m\le n$, on donne un encadrement de $r(n,m)$ valable pour $n^{2/3}\le m\le n$ en utilisant la relation $r(n,m)={\sum }_{t=1}^{\lfloor n/m\rfloor }P(n-(m-1)t,t)$ où $P(i,t)$ désigne le nombre de partitions de $i$ en exactement $t$ parts. On donne aussi une...

Given an odd prime number $p$, we characterize the partitions $\underline{\ell }$ of $p$ with $p$non negative parts ${\ell }_0\ge {\ell }_1\ge ...\ge {\ell }_{p-1}\ge 0$ for which there exist permutations $\sigma ,\tau$ of the set $\{0,...,p-1\}$ such that $p$ divides ${\sum }_{i=0}^{p-1}i{\ell }_{\sigma \left(i\right)}$ but does not divide ${\sum }_{i=0}^{p-1}i{\ell }_{\tau \left(i\right)}$. This happens if and only if the maximal number of equal parts of $\underline{\ell }$ is less than $p-2$. The question appeared when dealing with sums of $p$-th powers of resolvents, in order to solve a Galois module structure problem.

We demonstrate the Batyrev-Manin Conjecture for the number of points of bounded height on hypersurfaces of some toric varieties whose rank of the Picard group is 2. The method used is inspired by the one developed by Schindler for the case of hypersurfaces of biprojective spaces and by Blomer and Brüdern for some hypersurfaces of multiprojective spaces. These methods are based on the Hardy-Littlewood circle method. The constant obtained in the final asymptotic formula is the one conjectured by Peyre....

The aim of this survey article is to show certain questions concerning nuclear spaces and linear operators in normed spaces lead to questions from geometry of numbers.

For an odd prime p and an integer w ≥ 1, polynomial quotients $q_{p,w}\left(u\right)$ are defined by $q_{p,w}\left(u\right)\equiv (u^w-u^{wp})/pmodp$ with $0\le q_{p,w}\left(u\right)\le p-1$, u ≥ 0, which are generalizations of Fermat quotients $q_{p,p-1}\left(u\right)$. First, we estimate the number of elements $1\le u<N\le p$ for which $f\left(u\right)\equiv q_{p,w}\left(u\right)modp$ for a given polynomial f(x) over the finite field ${}_p$. In particular, for the case f(x)=x we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each element of...