Seminar za teoriju brojeva i algebru
lokacija:
PMF Matematički odsjek
vrijeme:
05.10.2016 - 10:15 - 12:00
U srijedu 5.10.2016. s početkom u 10:15, u predavaonici br. 104,
gosti seminara s Technische Universitat Graz, Austrija,
održat će tri kraća (30-minutna) predavanja:
Stefan Planitzer:
Romanoff's theorem and related problems
Kwok Chi Chim:
On a variant of Pillai's problem
Dijana Kreso:
Decomposable polynomials in second order linear recurrence sequences
Voditelji seminara:
Andrej Dujella
Ivica Gusić
Stefan Planitzer
Title: Romanoff's theorem and related problems
Abstract:
In 1934 Romanoff proved that the set of integers of the form $p+g^n$,
where $p$ is a prime, $n \in \mathbb{N}$ and $g\geq 2$ a fixed positive
integer, has positive lower density. The method of proof developed by
Romanoff was subsequently applied to a lot of problems of a similar
nature. We will give a short survey of some known classical and more
recent results concerning questions of Romanoff type and we will also
present some new results.
Kwok Chi Chim
Title: On a variant of Pillai's problem
Abstract:
In this talk I will present the results of a joint work with Istvan Pink
and Volker Ziegler on a variant of Pillai’s problem. Denote F_n and T_m
to be the n-th Fibonacci number and the m-th Tribonacci number
respectively. The problem is to find all integers c admitting at least
two representations of the form F_n – T_m for some positive integers
n > 1 and m > 1. With the use of linear forms in logarithms and a
generalized reduction method of Baker and Davenport, we proved that the
number of solutions for c is finite, and obtained all the values of c and
the corresponding representations in the form F_n – T_m. In this talk
I will give a highlight on the proof of the results.
Dijana Kreso
Title: Decomposable polynomials in second order linear recurrence sequences
Abstract:
In this talk I will present results that come from an ongoing joint work
with Clemens Fuchs and Christina Karolus from University of Salzburg.
We study elements of second order linear recurrence sequences
$(G_n)_{n=0}^{\infty}$ of polynomials in $\mathbb{C}[x]$ which are
decomposable, i.e. representable as $G_n=g\circ h$ for some
$g, h\in \mathbb{C}[x]$ satisfying $\deg g,\deg h>1$.
Under certain assumptions, and provided that $h$ is not of particular
type, we show that $\deg g$ may be bounded by a constant independent
of $n$, depending only on the sequence and the corresponding function
field. Our result resembles a result of Zannier from 2007 who showed that
if $f$ is a polynomial with $\ell$ non-constant terms and $f(x)=g(h(x))$,
and $h(x)$ is not of type $ax^k+b$, $a\neq 0$, then
$\deg g\leq 2\ell(\ell-1)$. The possible ways of writing a polynomial as
a composition of lower degree polynomials were studied by several authors,
starting with american mathematician Ritt in the 1920's. Results in this
area of mathematics have applications to various other areas, e.g. number
theory, complex analysis, etc. In my talk, I will present some Diophantine
applications.
