We obtain upper bound for the density of rational points on the cyclic covers of ${\mathbb{P}}^n$. As $n\to \infty$ our estimate tends to the conjectural bound of Serre.

A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly $k$ prime factors for $k>1$. Building upon a proof by E. M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree $n\le x$ with $k$ prime factors such that a fixed quadratic equation has exactly $2^k$ solutions modulo $n$.

Dati $m=2$ o $m=3$ numeri algebrici non nulli $\alpha =\left({\alpha }_1,\mathrm{\dots },{\alpha }_m\right)$ tali che ${\alpha }_j/{\alpha }_l$ non è una radice dell'unità per ogni $j\ne l$ , consideriamo una classe di determinanti di Vandermonde generalizzati di ordine quattro $G\left(a;x\right)$, al variare di $x$ in ${\mathbb{Z}}^4$ , connessa con alcuni problemi diofantei. Dimostriamo che il numero delle soluzioni $y\in {\mathbb{Z}}^3$ in posizione generica dell'equazione polinomiale-esponenziale disomogenea $G\left(a;0,y\right)=0$ non supera una costante esplicita $N\left(d\right)$ dipendente solo da $d=\left[\mathbb{Q}\right(\alpha {}_1,\mathrm{\dots },\alpha {}_m):\mathbb{Q}]$.

We discuss the distribution of Mordell-Weil ranks of the family of elliptic curves y² = (x + αf²)(x + βbg²)(x + γh²) where f,g,h are coprime polynomials that parametrize the projective smooth conic a² + b² = c² and α,β,γ are elements from ℚ̅. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.