C. Fuchs

activity

20142022

most citedDiophantine triples with values in

k

-generalized Fibonacci sequences

8 citations · 51 across the 20 of their papers we have counts for

collaborators

21 papers

math.NT2022★ 1 cited

On the growth of multi-recurrences

Clemens Fuchs, Sebastian Heintze

In this paper we provide a complete proof for a bound on the growth of multi-recurrences which are defined over a number field. The proven bound was already stated by van der Poort…

math.NT2021★ 2 cited

Approximation of values of algebraic elements over the ring of power sums

Clemens Fuchs, Sebastian Heintze

Let $Q E_{Z}$ be the set of power sums whose characteristic roots belong to $Z$ and whose coefficients belong to $Q$ , i.e. $ G :…

math.NT2020

Diophantine equations in separated variables and polynomial power sums

Clemens Fuchs, Sebastian Heintze

We consider Diophantine equations of the shape $f (x) = g (y)$ , where the polynomials $f$ and $g$ are elements of power sums. Using a finiteness criterion of Bilu and Tichy, we…

math.NT2020★ 6 cited

A function field variant of Pillai's problem

Clemens Fuchs, Sebastian Heintze

In this paper, we consider a variant of Pillai's problem over function fields $F$ in one variable over $C$ . For given simple linear recurrence sequences $G_{n}$ and $…

math.NT2020★ 5 cited

Integral zeros of a polynomial with linear recurrences as coefficients

Clemens Fuchs, Sebastian Heintze

Let $K$ be a number field, $S$ a finite set of places of $K$ , and $O_{S}$ be the ring of $S$ -integers. Moreover, let $$ G_n^{(0)} Z^d + \cdots + G_n^{(d-1)} Z + G…

math.NT2020★ 3 cited

A Polynomial Variant of Diophantine Triples in Linear Recurrences

Clemens Fuchs, Sebastian Heintze

Let $(G_{n})_{n = 0}^{\infty}$ be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation $ G_n = f_1α_1^n + \cdots + f_…

math.NT2020★ 4 cited

Norm form equations with solutions taking values in a multi-recurrence

Clemens Fuchs, Sebastian Heintze

We are interested in solutions of a norm form equation that takes values in a given multi-recurrence. We show that among the solutions there are only finitely many values in each c…

math.NT2020★ 6 cited

On the growth of linear recurrences in function fields

Clemens Fuchs, Sebastian Heintze

Let $(G_{n})_{n = 0}^{\infty}$ be a non-degenerate linear recurrence sequence with power sum representation $G_{n} = a_{1} (n) α_{1}^{n} + \dots + a_{t} (n) α_{t}^{n}$ . In this paper we will prov…

math.NT2019★ 3 cited

Perfect powers in polynomial power sums

Clemens Fuchs, Sebastian Heintze

We prove that a non-degenerate simple linear recurrence sequence $(G_{n} (x))_{n = 0}^{\infty}$ of polynomials satisfying some further conditions cannot contain arbitrary large powers…

math.NT2019

Yet another $S$ -unit variant of Diophantine tuples

Clemens Fuchs, Sebastian Heintze

We show that there are only finitely many triples of integers $0 < a < b < c$ such that the product of any two of them is the value of a given polynomial with integer coefficient…

math.NT2018★ 2 cited

Composite polynomials in linear recurrence sequences

Clemens Fuchs, Christina Karolus

Let $(G_{n} (x))_{n = 0}^{\infty}$ be a $d$ -th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover,…

math.NT2017★ 3 cited

Diophantine triples in linear recurrence sequences of Pisot type

Clemens Fuchs, Christoph Hutle, Florian Luca

The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in…

math.NT2017

Decomposable polynomials in second order linear recurrence sequences

Clemens Fuchs, Christina Karolus, Dijana Kreso

We study elements of second order linear recurrence sequences $(G_{n})_{n = 0}^{\infty}$ of polynomials in $C [x]$ which are decomposable, i.e. representable as $G_{n} = g \circ h$ …

math.NT2016★ 8 cited

Diophantine triples with values in $k$ -generalized Fibonacci sequences

Clemens Fuchs, Christoph Hutle, Florian Luca +1

We show that if $k \geq 2$ is an integer and $(F_{n}^{(k)})_{n \geq 0}$ is the sequence of $k$ -generalized Fibonacci numbers, then there are only finitely many triples of positive intege…

math.NT2015

30 years of collaboration

Clemens Fuchs, Lajos Hajdu

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Th…

math.NT2015

Only finitely many Tribonacci Diophantine triples exist

Clemens Fuchs, Christoph Hutle, Nurettin Irmak +2

Diophantine triples taking values in recurrence sequences have recently been studied quite a lot. In particular the question was raised whether or not there are finitely many Dioph…

math.NT2015★ 1 cited

The $p$ -adic analytic subgroup theorem revisited

Clemens Fuchs, Duc Hiep Pham

It is well-known that the Wüstholz' analytic subgroup theorem is one of the most powerful theorems in transcendence theory. The theorem gives in a very systematic and conceptual wa…

math.NT2014★ 5 cited

On fewnomials, integral points and a toric version of Bertini's theorem

Clemens Fuchs, Vincenzo Mantova, Umberto Zannier

An old conjecture of Erdős and Rényi, proved by Schinzel, predicted a bound for the number of terms of a polynomial $g (x) \in C [x]$ when its square $g (x)^{2}$ has a given nu…

math.NT2014

Elementary resolution of a family of quartic Thue equations over function fields

Clemens Fuchs, Ana Jurasić, Roland Paulin

We consider and completely solve the parametrized family of Thue equations \begin{eqnarray*}X(X-Y)(X+Y)(X-λY)+Y^4=ξ,\end{eqnarray*} where the solutions $x, y$ come from the ring $\m…

math.NT2014

Some applications of the $p$ -adic analytic subgroup theorem

Clemens Fuchs, Duc Hiep Pham

We use a $p$ -adic analogue of the analytic subgroup theorem of Wüstholz to deduce the transcendence and linear independence of some new classes of $p$ -adic numbers. In particular w…

math.NT2014★ 2 cited

Commutative algebraic groups and $p$ -adic linear forms

Clemens Fuchs, Duc Hiep Pham

Let $G$ be a commutative algebraic group defined over a number field $K$ that is disjoint over $K$ to $G_{a}$ and satisfies the condition of semistability. Consider a linear…

On the growth of multi-recurrences

Approximation of values of algebraic elements over the ring of power sums

Diophantine equations in separated variables and polynomial power sums

A function field variant of Pillai's problem

Integral zeros of a polynomial with linear recurrences as coefficients

A Polynomial Variant of Diophantine Triples in Linear Recurrences

Norm form equations with solutions taking values in a multi-recurrence

On the growth of linear recurrences in function fields

Perfect powers in polynomial power sums

Yet another S-unit variant of Diophantine tuples

Composite polynomials in linear recurrence sequences

Diophantine triples in linear recurrence sequences of Pisot type

Decomposable polynomials in second order linear recurrence sequences

Diophantine triples with values in k-generalized Fibonacci sequences

30 years of collaboration

Only finitely many Tribonacci Diophantine triples exist

The p-adic analytic subgroup theorem revisited

On fewnomials, integral points and a toric version of Bertini's theorem

Elementary resolution of a family of quartic Thue equations over function fields

Some applications of the p-adic analytic subgroup theorem

Commutative algebraic groups and p-adic linear forms

Yet another $S$ -unit variant of Diophantine tuples

Diophantine triples with values in $k$ -generalized Fibonacci sequences

The $p$ -adic analytic subgroup theorem revisited

Some applications of the $p$ -adic analytic subgroup theorem

Commutative algebraic groups and $p$ -adic linear forms