arXiv subjects

Claude Levesque

Publications and source records attributed to Claude Levesque.

11 recordsLinked to original sources

arXiv2018

Linear recurrence sequences and twisted binary forms

Let $ \prod_{i=1}^d (X-α_i Y) \in{\mathbb C}[X,Y]$ be a binary form and let $ε_1,\dots,ε_d$ be nonzero complex numbers. We consider the family of binary forms $ \prod_{i=1}^d (X-α_i ε_i^aY)$, $a\in {\mathbb Z}$, which we write as $$ X^d-U_1(a)X^{d-1}Y+\cdots+(-1)^{d-1} U_{d-1}(a) XY^{d-1}+(-1)^d U_d(a) Y^d.$$ In this paper we study these sequences $\bigl(U_h(a)\bigr)_{a\in {\mathbb Z}}$ which turn out to be linear recurrence sequences.

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arXiv2017

Representation of integers by cyclotomic binary forms

The homogeneous form $Φ_n(X,Y)$ of degree $φ(n)$ which is associated with the cyclotomic polynomial $ϕ_n(X)$ is dubbed a {\it cyclotomic binary form}. A positive integer $m\ge 1$ is said to be {\it representable by a cyclotomic binary form} if there exist integers $n,x,y$ with $n\ge 3$ and $\max\{|x|, |y|\}\ge 2$ such that $Φ_n(x,y)=m$. We prove that the number $a_m$ of such representations of $m$ by a cyclotomic binary form is finite. More precisely, we have $\,φ(n) \le ({2}/ {\log 3})\log m\, $ and $\, \max\{|x|,|y|\} \le ({2}/{\sqrt{3}})\, m^{1/φ(n)}.\,$ We give a description of the asymptotic cardinality of the set of values taken by the forms for $n\geq 3$. This will imply that the set of integers $m$ such that $a_m\neq 0$ has natural density 0. We will deduce that the average value of the integers $a_m$ among the nonzero values of $a_m$ grows like $\sqrt{\log \, m}$.

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arXiv2017

Families of Thue equations associated with a rank one subgroup of the unit group of a number field

Twisting a binary form $F_0(X,Y)\in{\mathbb{Z}}[X,Y]$ of degree $d\ge 3$ by powers $\upsilon^a$ ($a\in{\mathbb{Z}}$) of an algebraic unit $\upsilon$ gives rise to a binary form $F_a(X,Y)\in{\mathbb{Z}}[X,Y]$. More precisely, when $K$ is a number field of degree $d$, $σ_1,σ_2,\dots,σ_d$ the embeddings of $K$ into $\mathbb{C}$, $α$ a nonzero element in $K$, $a_0\in{\mathbb{Z}}$, $a_0>0$ and $$ F_0(X,Y)=a_0\displaystyle\prod_{i=1}^d (X-σ_i(α) Y), $$ then for $a\in{\mathbb{Z}}$ we set $$ F_a(X,Y)=\displaystyle a_0\prod_{i=1}^d (X-σ_i(α\upsilon^a) Y). $$ Given $m\ge 0$, our main result is an effective upper bound for the solutions $(x,y,a)\in{\mathbb{Z}}^3$ of the Diophantine inequalities $$ 0<|F_a(x,y)|\le m $$ for which $xy\not=0$ and ${\mathbb{Q}}(α\upsilon^a)=K$. Our estimate involves an effectively computable constant depending only on $d$; it is explicit in terms of $m$, in terms of the heights of $F_0$ and of $\upsilon$, and in terms of the regulator of the number field $K$.

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arXiv2015

Solving simultaneously Thue equations in the almost totally imaginary case

Let $α$ be an algebraic number of degree $d\ge 3$ having at most one real conjugate and let $K$ be the algebraic number field ${\mathbf Q}(α)$. For any unit $ε$ of $K$ such that ${\mathbf Q}(αε)=K$, we consider the irreducible polynomial $f_ε(X)\in{\mathbf Z}[X]$ such that $f_ε(αε)=0$. Let $F_ε(X,Y)\ = Y^df_ε(X/Y)\in{\mathbf Z}[X,Y]$ be the associated binary form. For each positive integer $m$, we exhibit an effectively computable bound for the solutions $(x,y,ε)$ of the diophantine equation $|F_ε(x,y)|\leq m$.

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arXiv2015

Familles d'équations de Thue associées à un sous-groupe de rang 1 d'unités totalement réelles d'un corps de nombres

Let $F$ be an irreducible binary form attached to a number field $K$ of degree $\geq 3$. Let $ε\not\in \{-1, 1\}$ be a totally real unit of $K$. By twisting $F$ with the powers $ε^a$ of $ε$, ($a\in{\mathbf Z}$), we obtain an infinite family $F_a$ of binary forms. Let $m\in{\mathbf Z}$. We give an effective bound for $\max\{|a|, \log|x|, \log|y|\}$ when $a,x,y$ are rational integers satisfying $F_a(x,y)=m$ with $xy\not=0$.

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arXiv2015

A family of Thue equations involving powers of units of the simplest cubic fields

E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms $F_n(X, Y )= X^3 -(n-1)X^2Y -(n+2)XY^2 -Y^3$ and the family of equations $F_n(X, Y )=\pm 1$, $n\in {\mathbf N}$. This family is associated to the family of the simplest cubic fields ${\mathbf Q}(λ)$ of D. Shanks, $λ$ being a root of $F_n(X,1)$. We introduce in this family a second parameter by replacing the roots of the minimal polynomial $F_n(X, 1) $ of $λ$ by the $a$-th powers of the roots and we effectively solve the family of Thue equations that we obtain and which depends now on the two parameters $n$ and $a$.

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arXiv2013

Some remarks on diophantine equations and diophantine approximation

We first recall the connection, going back to A. Thue, between rational approximation to algebraic numbers and integer solutions of some Diophantine equations. Next we recall the equivalence between several finiteness results on various Diophantine equations. We also give many equivalent statements of Mahler's generalization of the fundamental theorem of Thue. In particular, we show that the theorem of Thue--Mahler for degree $3$ implies the theorem of Thue--Mahler for arbitrary degree $\ge3$, and we relate it with a theorem of Siegel on the rational integral points of the projective line $¶^1(K)$ minus $3$ points. Finally we extend our study to higher dimensional spaces in connection with Schmidt's Subspace Theorem.

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arXiv2013

Familles d'equations de Thue-Mahler n'ayant que des solutions triviales

Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing the archimedean places and let $μ$, $α_1,α_2,α_3$ be non--zero elements in $K$. Denote by $\OS$ the ring of $S$--integers in $K$ and by $\OS^\times$ the group of $S$--units. Then the set of equivalence classes (namely, up to multiplication by $S$--units) of the solutions $(x,y,z,\varepsilon_1, \varepsilon_2,\varepsilon_3,\varepsilon)\in\OS^3\times(\OS^\times)^4$ of the diophantine equation $$ (X-α_1 E_1 Y) (X-α_2E_2 Y) (X-α_3E_3 Y)Z=μE, $$ satisfying $\Card\{α_1\varepsilon_1,α_2\varepsilon_2,α_3\varepsilon_3\}= 3$, is finite. With the help of this last result, we exhibit new families of Thue-Mahler equations having only trivial solutions. Furthermore, we produce an effective upper bound for the number of these solutions. The proofs of this paper rest heavily on Schmidt's subspace theorem.

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arXiv2013

Approximation of an algebraic number by products of rational numbers and units

We relate a previous result of ours on families of Diophantine equations having only trivial solutions with a result on the approximation of an algebraic number by products of rational numbers and units. We compare this approximation with a Liouville type estimate, and with an estimate arising from a lower bound for a linear combination of logarithms.

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arXiv2013

Families of cubic Thue equations with effective bounds for the solutions

To each non totally real cubic extension $K$ of $\Q$ and to each generator $α$ of the cubic field $K$, we attach a family of cubic Thue equations, indexed by the units of $K$, and we prove that this family of cubic Thue equations has only a finite number of integer solutions, by giving an effective upper bound for these solutions.

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arXiv2013

Solving effectively some families of Thue Diophantine equations

Let $α$ be an algebraic number of degree $d\ge 3$ and let $K$ be the algebraic number field $\Q(α)$. When $\varepsilon$ is a unit of $K$ such that $\Q(α\varepsilon)=K$, we consider the irreducible polynomial $f_\varepsilon(X) \in \Z[X]$ such that $f_\varepsilon(α\varepsilon)=0$. Let $F_\varepsilon(X,Y)$ be the irrreducible binary form of degree $d$ associated to $f_{\varepsilon}(X) $ under the condition $F_{\varepsilon}(X,1)=f_{\varepsilon}(X)$. For each positive integer $m$, we want to exhibit an effective upper bound for the solutions $(x,y,\varepsilon)$ of the diophantine inequation $|F_\varepsilon(x,y)|\le m$. We achieve this goal by restricting ourselves to a subset of units $\varepsilon$ which we prove to be sufficiently large as soon as the degree of $K$ is $\geq 4$.

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