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On some extensions of the Ailon-Rudnick Theorem

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Speaker: 
Alina Ostafe
Affiliation: 
UNSW
Date: 
Wed, 06/04/2016 - 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory downside up

Let $a,b$ be multiplicatively independent positive integers and
$\varepsilon>0$.  Bugeaud, Corvaja and Zannier (2003) proved that
$$
\gcd (a^n-1,b^n-1)\le \exp(\varepsilon n)
$$
for a sufficiently large $n$. Moreover, Ailon and Rudnick conjectured
that when $\gcd(a-1,b-1)=1$, then
$\gcd (a^n-1,b^n-1)=1$  infinitely often. Using finiteness of the number
of torsion points on curves, Ailon and Rudnick (2004) proved the
function field analogue of this conjecture, in a stronger form, that is,

Counting multiplicatively dependent vectors of algebraic numbers

Posted in
Speaker: 
Min Sha
Affiliation: 
UNSW
Date: 
Wed, 06/04/2016 - 16:30 - 17:30
Location: 
MPIM Lecture Hall
Parent event: 
Number theory downside up
Given a vector $v=(v_1, \ldots, v_n)$ of $n$ complex numbers, we say that $v$ is multiplicatively dependent if there is a non-zero integer vector $k=(k_1, \ldots, k_n)$ such that $v_1^{k_1} \cdots v_n^{k_n}=1$. In this talk, I will present some recent results on counting multiplicatively dependent vectors of algebraic numbers of fixed degree (or within a number field) and bounded height. These include sharp lower and upper bounds, and especially asymptotic formulas for several cases. (This is joint work with Francesco Pappalardi, Igor Shparlinski and Cameron Stewart.)

On coincidences among quadratic fields generated by the Shanks sequence

Posted in
Speaker: 
Igor Shparlinski
Affiliation: 
UNSW
Date: 
Thu, 07/04/2016 - 14:00 - 14:50
Location: 
MPIM Lecture Hall
Parent event: 
Number theory downside up
Motivated by work of Shanks (1969), we study the distribution of the fields ${\mathbb Q}\big(\sqrt{f(g^n)}\big)$ for a polynomial $f \in {\mathbb Z}[X]$ and an integer $g >1$. Using a variety of known results and techniques such as the abundance of shifted primes with a large prime divisor (R. Baker and G. Harman) and the square sieve (R.

Strategies to solve congruence problems

Posted in
Speaker: 
Ana Zumalacarregui
Affiliation: 
UNSW
Date: 
Thu, 07/04/2016 - 16:30 - 17:30
Location: 
MPIM Lecture Hall
Parent event: 
Number theory downside up

We will review some of the classical strategies to solve congruence problems and discuss the limits to them. We will focus in estimating the number of solutions to \[ f(x,y) \equiv 0 \pmod p \quad 1\le x,y \le M \] where $f$ is some interesting function (polynomial, exponential, etc.). When $M$ is large, the classical approach on character sums/Fourier Analysis allow us to obtain asymptotics for this quantity. Nevertheless, there seems to be a barrier to this method at $M=p^{1/2}$ and new ideas, based on Additive Combinatorics, are required for the case when $M$ is small.

 

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