We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $\ell$ of terms. Then we look at the possible decompositions $f\left(x\right)=g\left(h\right(x\left)\right)$, where $g,h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $\ell$ (and we provide explicit bounds). This supports and quantifies...

Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$. We prove Manin’s conjecture on the asymptotic (as $T\to \infty$) of the number of $K$-rational points of $X$ of height less than $T$, and give an explicit construction of a measure on $X\left(𝔸\right)$, generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆\left(K\right)$ on $X\left(𝔸\right)$. Our approach is based on the mixing property of $L^2(𝐆\left(K\right)\setminus 𝐆\left(𝔸\right))$ which we obtain with a rate of convergence. ...

We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $$\left\{\begin{array}{c}n=a_1+a_2+\cdots +a_{s-1},\hfill \\ a_1{a}_2\cdots a_{s-1}(a_1+a_2+\cdots +a_{s-1})=b^s\hfill \end{array}\right.$$ has positive integer or rational solutions $n$, $b$, $a_i$, $i=1,2,\cdots ,s-1$, $s\ge 3.$ Using the theory of elliptic curves, we prove that it has no positive integer solution for $s=3$, but there are infinitely many positive integers $n$ such that it has a positive integer solution for $s\ge 4$. As a corollary, for $s\ge 4$ and any positive integer $n$, the above Diophantine system has a positive rational solution. Meanwhile, we give conditions...

Consider a recurrence sequence ${\left(x_k\right)}_{k\in \mathbb{Z}}$ of integers satisfying $x_{k+n}=a_{n-1}{x}_{k+n-1}+...+a₁x_{k+1}+a₀x_k$, where $a₀,a₁,...,a_{n-1}\in \mathbb{Z}$ are fixed and a₀ ∈ -1,1. Assume that $x_k>0$ for all sufficiently large k. If there exists k₀∈ ℤ such that $x_{k₀}<0$ then for each negative integer -D there exist infinitely many rational primes q such that $q|x_k$ for some k ∈ ℕ and (-D/q) = -1.

In 1955, Roth established that if $\xi$ is an irrational number such that there are a positive real number $\epsilon$ and infinitely many rational numbers $p/q$ with $q\ge 1$ and $|\xi -p/q|\&lt;q^{-2-\epsilon }$, then $\xi$ is transcendental. A few years later, Cugiani obtained the same conclusion with $\epsilon$ replaced by a function $q\mapsto \epsilon \left(q\right)$ that decreases very slowly to zero, provided that the sequence of rational solutions to $|\xi -p/q|\&lt;q^{-2-\epsilon \left(q\right)}$ is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it...

Erdős space is the “rational” Hilbert space, that is, the set of vectors in ℓ² with all coordinates rational. Erdős proved that is one-dimensional and homeomorphic to its own square × , which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of . Let $M{ₙ}^{n+1}$, n ∈ ℕ, be the n-dimensional Menger continuum in ${ℝ}^{n+1}$, also known as the n-dimensional Sierpiński carpet, and let D be a countable dense subset of $M{ₙ}^{n+1}$. We consider the topological group...

In this note, we prove that there is no transcendental entire function $f\left(z\right)\in \mathbb{Q}\left[\right[z\left]\right]$ such that $f\left(\mathbb{Q}\right)\subseteq \mathbb{Q}$ and $\mathrm{den}f(p/q)=F\left(q\right)$, for all sufficiently large $q$, where $F\left(z\right)\in \mathbb{Z}\left[z\right]$.

Let $X$ be a uniruled projective manifold and let $x$ be a general point. The main result of [2] says that if the $(-K_X)$-degrees (i.e., the degrees with respect to the anti-canonical bundle of $X$) of all rational curves through $x$ are at least $dimX+1$, then $X$ is a projective space. In this paper, we study the structure of $X$ when the $(-K_X)$-degrees of all rational curves through $x$ are at least $dimX$. Our study uses the projective variety ${𝒞}_x\subset \mathbb{P}{T}_x\left(X\right)$, called the VMRT at $x$, defined as the union of tangent directions to the...

We consider two characteristic exponents of a rational function f:ℂ̂ → ℂ̂ of degree d ≥ 2. The exponent ${\chi }_a\left(f\right)$ is the average of log∥f’∥ with respect to the measure of maximal entropy. The exponent ${\chi }_m\left(f\right)$ can be defined as the maximal characteristic exponent over all periodic orbits of f. We prove that ${\chi }_a\left(f\right)={\chi }_m\left(f\right)$ if and only if f(z) is conformally conjugate to $z\mapsto z^{\pm d}$.

In this paper we introduce the concept of $\tau$-extending modules by $\tau$-rational submodules and study some properties of such modules. It is shown that the set of all $\tau$-rational left ideals of ${}_{R}R$ is a Gabriel filter. An $R$-module $M$ is called $\tau$-extending if every submodule of $M$ is $\tau$-rational in a direct summand of $M$. It is proved that $M$ is $\tau$-extending if and only if $M=Re{j}_{M}E(R/\tau \left(R\right))\oplus N$, such that $N$ is a $\tau$-extending submodule of $M$. An example is given to show that the direct sum of $\tau$-extending modules need not...

We study sums and products in a field. Let $F$ be a field with $\mathrm{ch}\left(F\right)\ne 2$, where $\mathrm{ch}\left(F\right)$ is the characteristic of $F$. For any integer $k\ge 4$, we show that any $x\in F$ can be written as $a_1+\cdots +a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1\cdots a_k=1$, and that for any $\alpha \in F\setminus \left\{0\right\}$ we can write every $x\in F$ as $a_1\cdots a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1+\cdots +a_k=\alpha$. We also prove that for any $x\in F$ and $k\in \{2,3,\cdots \}$ there are $a_1,\cdots ,a_{2k}\in F$ such that $a_1+\cdots +a_{2k}=x=a_1\cdots a_{2k}$.

Let ${\left(f_{\lambda }\right)}_{\lambda \in \Lambda }$ be a holomorphic family of rational mappings of degree $d$ on ${\mathbb{P}}^1\left(ℂ\right)$, with $k$ marked critical points $c_1,...,c_k$. To this data is associated a closed positive current $T_1\wedge \cdots \wedge T_k$ of bidegree $(k,k)$ on $\Lambda$, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which $c_1,...,c_k$ eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of $\mathrm{Supp}(T_1\wedge \cdots \wedge T_k)$. ...

This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve $C$ over $\mathbb{Q}$. The focus is on practical aspects of this problem in the case that the genus of $C$ is at least $2$, and therefore the set of rational points is finite.

We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in ${}_q\left[T\right]$ of degree d for which s consecutive coefficients $a=(a_{d-1},...,a_{d-s})$ are fixed. Our estimate asserts that $(d,s,a)={\mu }_{d}q+\left(q^{1/2}\right)$, where ${\mu }_d:={\sum }_{r=1}^d\left({(-1)}^{r-1}\right)/(r!)$. We also prove that $₂(d,s,a)=\mu {²}_{d}q²+\left(q^{3/2}\right)$, where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of ${}_q\left[T\right]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $₂(d,0)=\mu {²}_{d}q²+\left(q\right)$, where ₂(d,0) denotes the average second moment for...