This thesis is concerned with the problem of determining a measure of algebraic independence for a particular m-tuple θ₁,...,${\theta }_m$ of complex numbers. Specifically, let K be a number field and let f₁(z),...,$f_m\left(z\right)$ be elements of K[[z]] algebraically independent over K(z) satisfying equations of the form(*) $f_j\left(z^b\right)={\sum }_{i=1}^m{f}_i\left(z\right)a_{ij}\left(z\right)+b_j\left(z\right)$ (j = i,...,m)for b ≥ 2, $a_{ij}\left(z\right)$, $b_j\left(z\right)$ in K(z). Suppose finally that α ∈ K is such that 0 < |α| < 1, the $f_j(z$) converge at z = α and the $a_{ij}\left(z\right)$, $b_j\left(z\right)$ are analytic at $z=\alpha ,{\alpha }^b,{\alpha }^{b²},...$ Then the ${\theta }_i=f_i\left(\alpha \right)$ are algebraically independent...

Let d be a positive integer and α a real algebraic number of degree d + 1. Set $\alpha ̲:=(\alpha ,\alpha ²,...,{\alpha }^d)$. It is well-known that $c\left(\alpha ̲\right):=limin{f}_{q\to \infty }{q}^{1/d}·\left|\right|q\alpha ̲\left|\right|>0$, where ||·|| denotes the distance to the nearest integer. Furthermore, $c\left(\alpha ̲\right)n^{-1/d}\le c\left(n\alpha ̲\right)\le nc\left(\alpha ̲\right)$ for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that $c\left(n\alpha ̲\right)\le C{n}^{-1/d}$ for any integer n ≥ 1.

We consider the Tribonacci sequence $T:={T_n}_{n\ge 0}$ given by T₀ = 0, T₁ = T₂ = 1 and $T_{n+3}=T_{n+2}+T_{n+1}+T_n$ for all n ≥ 0, and we find all triples of Tribonacci numbers which are multiplicatively dependent.

Very recently, the generating function $A\left(z\right)$ of the Stern sequence ${\left(a_n\right)}_{n\ge 0}$, defined by $a_0:=0,a_1:=1,$ and $a_{2n}:=a_n,a_{2n+1}:=a_n+a_{n+1}$ for any integer $n\&gt;0$, has been considered from the arithmetical point of view. Coons [8] proved the transcendence of $A\left(\alpha \right)$ for every algebraic $\alpha$ with $0\&lt;\left|\alpha \right|\&lt;1$, and this result was generalized in [6] to the effect that, for the same $\alpha$’s, all numbers $A\left(\alpha \right),A^{\prime }\left(\alpha \right),A^{\prime \prime }\left(\alpha \right),...$ are algebraically independent. At about the same time, Bacher...

Let $𝒜$ be a commutative algebraic group defined over a number field $k$. We consider the following question:A complete answer for the case of the multiplicative group ${𝔾}_m$ is classical. We study other instances and in particular obtain an affirmative answer when $r$ is a prime and $𝒜$ is either an elliptic curve or a torus of small dimension with respect to $r$. Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when $r$ is a prime. ...

Let $𝒱$ be a fixed algebraic variety defined by $m$ polynomials in $n$ variables with integer coefficients. We show that there exists a constant $C\left(𝒱\right)$ such that for almost all primes $p$ for all but at most $C\left(𝒱\right)$ points on the reduction of $𝒱$ modulo $p$ at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

Let $p=1+2^{e+1}q$ be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order $2^{e+1}$ (resp. order $d_{\phi }$ dividing q), and let ψₙ be an even character of conductor $p^{n+1}$ and order pⁿ. We put χ = δφψₙ, whose value is contained in $Kₙ=\mathbb{Q}\left({\zeta }_{(p-1)pⁿ}\right)$. It is well known that the Bernoulli number $B_{1,\chi }$ is not zero, which is shown in an analytic way. In the extreme cases $d_{\phi }=1$ and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: $T{r}_{n/1}\left(\xi B_{1,\chi }\right)\ne 0$ for any...

We study the compositum $k^{\left[d\right]}$ of all degree $d$ extensions of a number field $k$ in a fixed algebraic closure. We show $k^{\left[d\right]}$ contains all subextensions of degree less than $d$ if and only if $d\le 4$. We prove that for $d\&gt;2$ there is no bound $c=c\left(d\right)$ on the degree of elements required to generate finite subextensions of $k^{\left[d\right]}/k$. Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of $d$, but that one can take $c=d$ when $d$ is prime. This question was inspired by work of...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

Let $G$ be a group and $\omega \left(G\right)$ be the set of element orders of $G$. Let $k\in \omega \left(G\right)$ and $m_k\left(G\right)$ be the number of elements of order $k$ in $G$. Let nse$\left(G\right)=\{m_k\left(G\right):k\in \omega \left(G\right)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$\left(G\right)=$ nse$\left(S_r\right)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

Let K be an algebraic number field with non-trivial class group G and ${}_K$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_k\left(x\right)$ denote the number of non-zero principal ideals $a_K$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that $F_k\left(x\right)$ behaves, for x → ∞, asymptotically like $x{\left(logx\right)}^{1/\left|G\right|-1}{\left(loglogx\right)}^{{}_k\left(G\right)}$. In this article, it is proved that for every prime p, $₁\left(C_p\oplus C_p\right)=2p$, and it is also proved that $₁\left(C_{mp}\oplus C_{mp}\right)=2mp$ if $₁\left(C_m\oplus C_m\right)=2m$ and m is large enough. In particular, it is shown...

In this paper, we examine a natural question concerning the divisors of the polynomial $x^n-1$: “How often does $x^n-1$ have a divisor of every degree between $1$ and $n$?” In a previous paper, we considered the situation when $x^n-1$ is factored in $\mathbb{Z}\left[x\right]$. In this paper, we replace $\mathbb{Z}\left[x\right]$ with ${𝔽}_p\left[x\right]$, where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$.

Let $f:{\mathbb{P}}^k\to {\mathbb{P}}^k$ be a dominating rational mapping of first algebraic degree $\lambda \ge 2$. If $S$ is a positive closed current of bidegree $(1,1)$ on ${\mathbb{P}}^k$ with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks ${\lambda }^{-n}{\left(f^n\right)}^{*}S$ converge to the Green current $T_f$. For some families of mappings, we get finer convergence results which allow us to characterize all $f^{*}$-invariant currents.

A cluster ensemble is a pair $(𝒳,𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $\Gamma$. The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p:𝒜\to 𝒳$. The space $𝒜$ is equipped with a closed $2$-form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...