Forniamo un calcolo esplicito della funzione di partizione di Kostant per algebre di Lie complesse di rango $2$. La tecnica principale consiste nella riduzione a casi più semplici ed all'uso di funzioni generatrici.

Dans cette note nous décrivons différentes méthodes utilisées en pratique pour calculer le nombre de classes d'un corps quadratique imaginaire ou réel ainsi que pour calculer le régulateur d'un corps quadratique réel. En particulier nous décrivons l'infrastructure de Shanks ainsi que la méthode sous-exponentielle de McCurley.

Nous décrivons dans cet article les algorithmes nécessaires à une implantation efficace de la méthode de Schoof pour le calcul du nombre de points sur une courbe elliptique dans un corps fini. Nous tentons d’unifier pour cela les idées d’Atkin et d’Elkies. En particulier, nous décrivons le calcul d’équations pour $X_0\left(\ell \right)$, $\ell$ premier, ainsi que le calcul efficace de facteurs des polynômes de division d’une courbe elliptique.

L’article comporte une méthode de calcul de fonctions de Belyi “optimales”, associées à des dessins plans. Cette étude conduit à s’interroger sur la possibilité de définir une fonction de Belyi sur le corps des modules du dessin. Pour les arbres par exemple, nous montrons que c’est toujours le cas. La preuve donne une méthode pour spécifier une telle fonction. Nous donnons ensuite un exemple de dessin qui n’admet pas de fonction de Belyi sur son corps des modules. Enfin, nous étudions la question...

Let $h$ be a complex valued multiplicative function. For any $N\in \mathbb{N}$, we compute the value of the determinant $D_N:={det}_{i|N,j|N}\left(\frac{h\left(\left(i,j\right)\right)}{ij}\right)$ where $\left(i,j\right)$ denotes the greatest common divisor of $i$ and $j$, which appear in increasing order in rows and columns. Precisely we prove that $$D_N=\prod _{p{}^l\parallel N}{\left(\frac{1}{p^{l\left(l+1\right)}}\stackrel{l}{\prod _{i=1}}\left(h\left(p^i\right)-h\left(p^{i-1}\right)\right)\right)}^{\tau \left(N/p^l\right)}.$$ This means that $D_N^{1/\tau \left(N\right)}$ is a multiplicative function of $N$. The algebraic apparatus associated with this result allows us to prove the following two results. The first one is the characterization of real multiplicative functions $f\left(n\right)$, with $0\le f\left(p\right)<1$, as minimal values of certain...

It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $\xi$ of the polynomial $P_t\left(x\right)=x^4-t{x}^3-6x^2+tx+1$, assuming that $t>0$, $t\ne 3$ and $t^2+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal...

The coefficients of the greatest common divisor of two polynomials $f$ and $g$ (GCD$(f,g)$) can be obtained from the Sylvester subresultant matrix $S_j(f,g)$ transformed to lower triangular form, where $1\le j\le d$ and $d=$ deg(GCD$(f,g)$) needs to be computed. Firstly, it is supposed that the coefficients of polynomials are given exactly. Transformations of $S_j(f,g)$ for an arbitrary allowable $j$ are in details described and an algorithm for the calculation of the GCD$(f,g)$ is formulated. If inexact polynomials are given, then an approximate greatest...

Let $R$ be a non-maximal order in a finite algebraic number field with integral closure $\overline{R}$. Although $R$ is not a unique factorization domain, we obtain a positive integer $N$ and a family $𝒬$ (called a Cale basis) of primary irreducible elements of $R$ such that $x^N$ has a unique factorization into elements of $𝒬$ for each $x\in R$ coprime with the conductor of $R$. Moreover, this property holds for each nonzero $x\in R$ when the natural map $\mathrm{Spec}\left(\overline{R}\right)\to \mathrm{Spec}\left(R\right)$ is bijective. This last condition is actually equivalent to several properties linked...

We give the answer to the question in the title by proving that $$L_{18}=5778=5555+222+1$$ is the largest Lucas number expressible as a sum of exactly three repdigits. Therefore, there are many Lucas numbers which are sums of three repdigits.