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Search by event  2256 | enregistrements trouvés : 28

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A Diophantine equation has separated variables if it is of the form $ f(x) = g(y)$ for polynomials $f$, $g$. In a more general sense the degree of $f $ and $g$ may also be a variable.In the present talk various results for special types of the polynomials $f$ and $g$ will be presented. The types of the considered polynomials contain power sums, sums of products of consecutive integers, Komornik polynomials, perfect powers. Results on $F$-Diophantine sets, which are proved using results on Diophantine equations in separated variables will also be considered. The main tool for the proof of the presented general qualitative results is the famous Bilu-Tichy Theorem. Further, effective results (which depend on Baker’s method) and results containing the complete solutions to special cases of these equations will also be included.
A Diophantine equation has separated variables if it is of the form $ f(x) = g(y)$ for polynomials $f$, $g$. In a more general sense the degree of $f $ and $g$ may also be a variable.In the present talk various results for special types of the polynomials $f$ and $g$ will be presented. The types of the considered polynomials contain power sums, sums of products of consecutive integers, Komornik polynomials, perfect powers. Results on $F...

11D41 ; 11C08

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We will discuss a new type of ergodic theorem which has among its corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg and a theorem of Erdös-Delange. This ergodic approach leads to a new dynamical framework for a general form of Sarnak’s Möbius disjointness conjecture which focuses on the "joint independence" of actions of (N,+) and (N,x). The talk is based on recent joint work with Florian Richter.
We will discuss a new type of ergodic theorem which has among its corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg and a theorem of Erdös-Delange. This ergodic approach leads to a new dynamical framework for a general form of Sarnak’s Möbius disjointness conjecture which focuses on the "joint independence" of actions of (N,+) and (N,x). The talk is based on ...

37A45 ; 11N99 ; 11J71

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A cap is a point set in affine or projective space without any three points on any line. We will discuss the current state of
the art, and give an exponential improvement for the size of caps of AG(n, p), which one can think of as (Z/pZ)^n, and PG(n,p). For certain primes, 5,11,17,23,29 and 41, we improve the asymptotic growth of these caps, for example, when p=23 from (8.091...)^n to (9-o(1))^n, as n tends to infinity.

51E20 ; 51E22 ; 05B25

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Let $q=p^r$, where $p$ is a prime number and $ ß=(\beta_1 ,\ldots ,\beta_r)$ be a basis of $\mathbb{F}_q$ over $\mathbb{F}_p$.
Any $\xi \in \mathbb{F}_q$ has a unique representation $\xi =\sum_{i=1}^r x_i \beta _i$ with $x_1,\ldots ,x_r \in \mathbb{F}_p$.
The coefficients $x_1,\ldots ,x_r$ are called the digits of $\xi$ with respect to the basis $ß$.
The analog of the Rudin-Shapiro function is $R(\xi)=x_1x_2+\cdots + x_{r-1}x_r$. For $f \in \mathbb{F}_q [X]$, non constant and $c\in\mathbb{F}_p$, we obtain some formulas for the number of solutions in $\mathbb{F}_q$ of $R(f(\xi ))=c$. The proof uses the Hooley-Katz bound for the number of zeros of polynomials in $\mathbb{F}_p$ with several variables.

This is a joint work with László Mérai and Arne Winterhof.
Let $q=p^r$, where $p$ is a prime number and $ ß=(\beta_1 ,\ldots ,\beta_r)$ be a basis of $\mathbb{F}_q$ over $\mathbb{F}_p$.
Any $\xi \in \mathbb{F}_q$ has a unique representation $\xi =\sum_{i=1}^r x_i \beta _i$ with $x_1,\ldots ,x_r \in \mathbb{F}_p$.
The coefficients $x_1,\ldots ,x_r$ are called the digits of $\xi$ with respect to the basis $ß$.
The analog of the Rudin-Shapiro function is $R(\xi)=x_1x_2+\cdots + x_{r-1}x_r$. For $f \in ...

11A63 ; 11T23 ; 11T30

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This is the second part of the talk of Daniel Fiorilli. We will explain the proofs of our theorem about the moments of moments of primes in arithmetic progressions.

11N05 ; 11M26 ; 11N13

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This talk will report on a work with S. Bettin (University of Genova) in which we obtained exact modularity relations for the q-Pochhammer symbol, which is a finite version of the Dedekind eta function. We will overview some of their useful aspects and applications, in particular to the value distribution of a certain knot invariants, the Kashaev invariants, constructed with q-Pochhammer symbols.

11B65 ; 57M27 ; 11F03 ; 60F05

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It is well known that the every letter $\alpha$ of an automatic sequence $a(n)$ has a logarithmic density -- and it can be decided when this logarithmic density is actually adensity. For example, the letters $0$ and $1$ of the Thue-Morse sequences $t(n)$ have both frequences $1/2$. The purpose of this talk is to present a corresponding result for subsequences of general automatic sequences along primes and squares. This is a far reaching of two breakthroughresults of Mauduit and Rivat from 2009 and 2010, where they solved two conjectures by Gelfond on the densities of $0$ and $1$ of $t(p_n)$ and $t(n^2)$ (where $p_n$ denotes thesequence of primes). More technically, one has to develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. Then asan application one can deduce that the logarithmic densities of any automatic sequence along squares $(n^2){n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, if densities exist then they are (usually) rational.
It is well known that the every letter $\alpha$ of an automatic sequence $a(n)$ has a logarithmic density -- and it can be decided when this logarithmic density is actually adensity. For example, the letters $0$ and $1$ of the Thue-Morse sequences $t(n)$ have both frequences $1/2$. The purpose of this talk is to present a corresponding result for subsequences of general automatic sequences along primes and squares. This is a far reaching of two ...

11B85 ; 11L20 ; 11N05 ; 11A63 ; 11L03

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Virtualconference  $D(n)$-sets with square elements
Dujella, Andrej (Auteur de la Conférence) | CIRM (Editeur )

For an integer n, a set of distinct nonzero integers $\left \{ a_{1},a_{2},...a_{m} \right \}$ such that $a_{i}a_{j}+n$ is a perfect square for all 1 ≤ i < j ≤ m, is called a Diophantine m-tuple with the property $D(n)$ or simply a $D(n)$-set. $D(1)$-sets are known as Diophantine m-tuples. When considering $D(n)$-sets, usually an integer n is fixed in advance. However, we may ask if a set can have the property $D(n)$ for several different n’s. For example, {8, 21, 55} is a $D(1)$-triple and $D(4321)$-triple. In a joint work with Adzaga, Kreso and Tadic, we presented several families of Diophantine triples which are $D(n)$-sets for two distinct n’s with $n\neq 1$. In a joint work with Petricevic we proved that there are infinitely many (essentially different) quadruples which are simultaneously $D(n_{1})$-quadruples and $D(n_{2})$-quadruples with $n_{1}\neq n_{2}$. Morever, the elements in some of these quadruples are squares, so they are also $D(0)$-quadruples. E.g. $\left \{ 54^{2}, 100^{2}, 168^{2}, 364^{2}\right \} $ is a $D(8190^{2})$, $D(40320^{2})$ and $D(0)$-quadruple. In this talk, we will describe methods used in constructions of mentioned triples and quadruples. We will also mention a work in progress with Kazalicki and Petricevic on $D(n)$-quintuples with square elements (so they are also $D(0)$-quintuples). There are infinitely many such quintuples. One example is a $D(4804802)$-quintuple $\left \{ 225^{2}, 286^{2}, 819^{2}, 1408^{2}, 2548^{2}\right \}$.
For an integer n, a set of distinct nonzero integers $\left \{ a_{1},a_{2},...a_{m} \right \}$ such that $a_{i}a_{j}+n$ is a perfect square for all 1 ≤ i < j ≤ m, is called a Diophantine m-tuple with the property $D(n)$ or simply a $D(n)$-set. $D(1)$-sets are known as Diophantine m-tuples. When considering $D(n)$-sets, usually an integer n is fixed in advance. However, we may ask if a set can have the property $D(n)$ for several different n’s. ...

11D09 ; 11G05 ; 11Y50

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Let $K$ be a number field, $\alpha _1,...,\alpha _t \in K$ and $G$ a finite abelian group. We explain how to construct explicitly a normal extension $L$ of $K$ with Galois group $G$, such that all of the elements $\alpha_{i}$ are norms of elements of $L$. The construction is based on class field theory and a recent formulation of Tate’s criterion for the validity of the Hasse norm principle. This is joint work with Rodolphe Richard (UCL).

11Y40 ; 11R37 ; 14G05 ; 11D57

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In the 1980’s we developed an effective specialization method and used it to prove effective finiteness theorems for Thue equations, decomposable form equations and discriminant equations over a restricted class of finitely generated domains (FGD’s) over $\mathbb{Z}$ which may contain not only algebraic but also transcendental elements. In 2013 we refined with Evertse the method and combined it with an effective result of Aschenbrenner (2004) concerning ideal membership in polynomial rings over $\mathbb{Z}$ to establish effective results over arbitrary FGD’s over $\mathbb{Z}$. By means of our method general effective finiteness theorems have been obtained in quantitative form for several classical Diophantine equations over arbitrary FGD’s, including unit equations, discriminant equations (Evertse and Gyory, 2013, 2017), Thue equations, hyper- and superelliptic equations, the Schinzel-Tijdeman equation (Bérczes, Evertse and Gyory, 2014), generalized unit equations (Bérczes, 2015), and the Catalan equation (Koymans, 2015). In the first part of the talk we shall briefly survey these results. Recently we proved with Evertse effective finiteness theorems in quantitative form for norm form equations, discriminant form equations and more generally for decomposable form equations over arbitrary FGD’s. In the second part, these new results will be presented. Some applications will also be discussed.
In the 1980’s we developed an effective specialization method and used it to prove effective finiteness theorems for Thue equations, decomposable form equations and discriminant equations over a restricted class of finitely generated domains (FGD’s) over $\mathbb{Z}$ which may contain not only algebraic but also transcendental elements. In 2013 we refined with Evertse the method and combined it with an effective result of Aschenbrenner (2004) ...

11D57 ; 11D61 ; 11D72

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Roots of unity of order dividing $n$ equidistribute around the unit circle as $n$ tends to infinity. With some extraeffort the same can be shown when restricting to roots of unity of exact order $n$. Equidistribution is measured by comparing the average of a continuous test function evaluated at these roots of unity with the integral over the complex unit circle. Baker, Ih, and Rumely extended this to test function with logarithmic singularities of the form $\log|P|$ where $P$ is a univariate polynomial in algebraic coefficients. I will discuss joint work with Vesselin Dimitrov where we allow $P$ to come from a class of a multivariate polynomials, extending a result of Lind, Schmidt, and Verbitskiy. Our method draws from earlier work of Duke.
Roots of unity of order dividing $n$ equidistribute around the unit circle as $n$ tends to infinity. With some extraeffort the same can be shown when restricting to roots of unity of exact order $n$. Equidistribution is measured by comparing the average of a continuous test function evaluated at these roots of unity with the integral over the complex unit circle. Baker, Ih, and Rumely extended this to test function with logarithmic singularities ...

11J83 ; 11R06 ; 14G40 ; 37A45 ; 37P30

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Exponential Diophantine equations, say of the form (1) $u_{1}+...+u_{k}=b$ where the $u_{i}$ are exponential terms with fixed integer bases and unknown exponents and b is a fixed integer, play a central role in the theory of Diophantine equations, with several applications of many types. However, we can bound the solutions only in case of k = 2 (by results of Gyory and others, based upon Baker’s method), for k > 2 only the number of so-called non-degenerate solutions can be bounded (by the Thue-Siegel-Roth-Schmidt method; see also results of Evertse and others). In particular, there is a big need for a method which is capable to solve (1) completely in concrete cases.
Skolem’s conjecture (roughly) says that if (1) has no solutions, then it has no solutions modulo m with some m. In the talk we present a new method which relies on the principle behind the conjecture, and which (at least in principle) is capable to solve equations of type (1), for any value of k. We give several applications, as well. Then we provide results towards the solution of Skolem’s conjecture. First we show that in certain sense it is ’almost always’ valid. Then we provide a proof for the conjecture in some cases with k = 2, 3. (The handled cases include Catalan’s equation and Fermat’s equation, too - the precise connection will be explained in the talk). Note that previously Skolem’s conjecture was proved only for k = 1, by Schinzel.
The new results presented are (partly) joint with Bertok, Berczes, Luca, Tijdeman.
Exponential Diophantine equations, say of the form (1) $u_{1}+...+u_{k}=b$ where the $u_{i}$ are exponential terms with fixed integer bases and unknown exponents and b is a fixed integer, play a central role in the theory of Diophantine equations, with several applications of many types. However, we can bound the solutions only in case of k = 2 (by results of Gyory and others, based upon Baker’s method), for k > 2 only the number of so-called ...

11D41 ; 11D61 ; 11D79

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Virtualconference  Fibonacci numbers and repdigits
Luca, Florian (Auteur de la Conférence) | CIRM (Editeur )

In the first part of the talk we will survey known results concerning Fibonacci numbers whose digital representations in base 10 display some interesting patterns. In the second part of the talk we will give the main steps of the proof of a recent result which states that $b = 4$ is the only integer ≥ 2 such that there are two Fibonacci numbers larger than 1 which are repunits in base b. In this case, $F_{5}=(4^{2}-1)/(4-1)$ and $F_{8}=(4^{3}-1)/(4-1)$. This is joint work with C. A. Gomez and J. C. Gomez from Cali, Colombia.
In the first part of the talk we will survey known results concerning Fibonacci numbers whose digital representations in base 10 display some interesting patterns. In the second part of the talk we will give the main steps of the proof of a recent result which states that $b = 4$ is the only integer ≥ 2 such that there are two Fibonacci numbers larger than 1 which are repunits in base b. In this case, $F_{5}=(4^{2}-1)/(4-1)$ and $F_{8}=...

11A63 ; 11B39 ; 11D61

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This is joint work with Jörg Thuswaldner from University of Leoben.

A linear recurrent number system is a generalization of the $q$-adic number system, where we replace the sequence of powers of $q$ by a linear recurrent sequence $G_{k+d}=a_1G_{k+d-1}+\cdots+a_dG_k$ for $k\geq 0$. Under some mild conditions on the recurrent sequence every positive integer $n$ has a representation of the form \[n=\sum_{j=0}^k \varepsilon_j(n)G_j.\]

The $q$-adic number system corresponds to the linear recursion $G_{k+1}=qG_k$ and $G_0=1$. The first example of a real generalization is due to Zeckendorf who showed that the Fibonacci sequence $G_0=1$, $G_1=2$, $G_{k+2}=G_{k+1}+G_k$ for $k\geq0$ yields a representation for each positive integer. This is unique if we additionally suppose that no two consecutive ones exist in the representation. Similar restrictions hold for different recurrent sequences and they build the essence of these number systems.

In the present talk we investigate the representation of primes and almost primes in linear recurrent number systems. We start by showing the different results due to Fouvry, Mauduit and Rivat in the case of $q$-adic number systems. Then we shed some light on their main tools and techniques. The heart of our considerations is the following Bombieri-Vinogradov type result
\[\sum_{q < x^{\vartheta-\varepsilon}}\max_{y < x}\max_{1\leq a\leq q} \left\vert\sum_{\substack{n< y,s_G(n)\equiv b\bmod d\\ n\equiv b\bmod q}}1 -\frac1q\sum_{n < y,s_G(n)\equiv b\bmod d}1\right\vert \ll x(\log 2x)^{-A},\]
which we establish under the assumption that $a_1\geq30$. This lower bound is due to numerical estimations. With this tool in hand we are able to show that \[ \left\vert\{n\leq x\colon s_G(n)\equiv b\bmod d, n=p_1\text{ or }n=p_1p_2\}\right\vert\gg \frac{x}{\log x}.\]
This is joint work with Jörg Thuswaldner from University of Leoben.

A linear recurrent number system is a generalization of the $q$-adic number system, where we replace the sequence of powers of $q$ by a linear recurrent sequence $G_{k+d}=a_1G_{k+d-1}+\cdots+a_dG_k$ for $k\geq 0$. Under some mild conditions on the recurrent sequence every positive integer $n$ has a representation of the form \[n=\sum_{j=0}^k \varepsilon_j(n)G_j.\]

The $q$-adic ...

11A63 ; 11L07 ; 11N05

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Some number-theoretic problems have led to the study of some infinite series that show multifractal behaviour, which means that their Hölder pointwise regularity may widely change from point to point. Reviewing some examples such as lacunary trigonometric series, Davenport series, Brjuno-type functions, I will put emphasis on the methods encountered in the literature to compute the pointwise Hölder exponent of such functions.

11A55 ; 26A15 ; 26A30 ; 28A80

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We will discuss recent progress in analysis of uniform and ordinary Diophantine exponents $\hat\omega $ and $\omega$ for linear Diophantine approximation as well as some applications of the related methods. In particular, we give a new criterion for badly approximable vectors in $\mathbb{R}^{d}$ the behavior of the best approximation vectors in the sense of simultaneous approximation and in the sense of linear form. It turned out that compared to the one-dimensional case our criterion is rather unusual. We apply this criterion to the analysis of Dirichlet spectrum for simultaneous Diophantine approximation.
We will discuss recent progress in analysis of uniform and ordinary Diophantine exponents $\hat\omega $ and $\omega$ for linear Diophantine approximation as well as some applications of the related methods. In particular, we give a new criterion for badly approximable vectors in $\mathbb{R}^{d}$ the behavior of the best approximation vectors in the sense of simultaneous approximation and in the sense of linear form. It turned out that compared ...

11J13 ; 11J06 ; 11J70

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In this talk we look at polynomials having the property that all compositional iterates are irreducible, which we call dynamical irreducible. After surveying some previous results (mostly over finite fields), we will concentrate on the question of the dynamical irreducibility of integer polynomials being preserved in reduction modulo primes. More precisely, for a class of integer polynomials $f$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions, he set of primes $p$ such that $f$ is dynamical irreducible modulo $p$ is of relative density zero. The proof of this result relies on a combination of analytic (the square sieve) and diophantine (finiteness of solutions to certain hyperelliptic equations) tools, which we will briefly describe.
In this talk we look at polynomials having the property that all compositional iterates are irreducible, which we call dynamical irreducible. After surveying some previous results (mostly over finite fields), we will concentrate on the question of the dynamical irreducibility of integer polynomials being preserved in reduction modulo primes. More precisely, for a class of integer polynomials $f$, which in particular includes all quadratic ...

11R09 ; 11R45 ; 11L40 ; 37P25

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Virtualconference  Bertini and Northcott
Pazuki, Fabien (Auteur de la Conférence) | CIRM (Editeur )

I will report on joint work with Martin Widmer. Let X be a smooth projective variety over a number field K. We prove a Bertini-type theorem with explicit control of the genus, degree, height, and field of definition of the constructed curve on X. As a consequence we provide a general strategy to reduce certain height and rank estimates on abelian varieties over a number field K to the case of Jacobian varieties defined over a suitable extension of K. We will give examples where the strategy works well!
I will report on joint work with Martin Widmer. Let X be a smooth projective variety over a number field K. We prove a Bertini-type theorem with explicit control of the genus, degree, height, and field of definition of the constructed curve on X. As a consequence we provide a general strategy to reduce certain height and rank estimates on abelian varieties over a number field K to the case of Jacobian varieties defined over a suitable extension ...

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For any fixed coprime positive integers a, b and c with min{a, b, c} > 1, we prove that the equation $a^{x}+b^{y}=c^{z}$ has at most two solutions in positive integers x, y and z, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of Fermat’s equation, it is also regarded as a 3-variable generalization of the celebrated theorem of Bennett [M.A.Bennett, On some exponential equations of S.S.Pillai, Canad. J. Math. 53(2001), no.2, 897-922] which asserts that Pillai’s type equation $a^{x}-b^{y}=c$ has at most two solutions in positive integers x and y for any fixed positive integers a, b and c with min {a, b} > 1. In this talk we give a brief summary of corresponding earlier results and present the main improvements leading to this definitive result. This is a joint work with T. Miyazaki.
For any fixed coprime positive integers a, b and c with min{a, b, c} > 1, we prove that the equation $a^{x}+b^{y}=c^{z}$ has at most two solutions in positive integers x, y and z, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of ...

11D61 ; 11D41 ; 11D45

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In the lecture we prove a lower estimate for the average of the absolute value of the remainder term of the prime number theorem which depends in an explicit way on a given zero of the Riemann Zeta Function. The estimate is only interesting if this hypothetical zero lies off the critical line which naturally implies the falsity of the Riemann Hypothesis. (If the Riemann Hypothesis is true, stronger results areobtainable by other metods.) The first explicit results in this direction were proved by Turán and Knapowski in the 1950s, answering a problem of Littlewood from the year 1937. They used the power sum method of Turán. Our present approach does not use Turán’s method and gives sharper results.
In the lecture we prove a lower estimate for the average of the absolute value of the remainder term of the prime number theorem which depends in an explicit way on a given zero of the Riemann Zeta Function. The estimate is only interesting if this hypothetical zero lies off the critical line which naturally implies the falsity of the Riemann Hypothesis. (If the Riemann Hypothesis is true, stronger results areobtainable by other metods.) The ...

11M26 ; 11N05 ; 11N30

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