arXiv subjects

Marley Young

Publications and source records attributed to Marley Young.

6 recordsLinked to original sources

arXiv2024

On multiplicatively dependent vectors of polynomial values

Given polynomials $f_1,\ldots,f_n$ in $m$ variables with integral coefficients, we give upper bounds for the number of integral $m$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_n$ of bounded height such that $f_1(\mathbf{u}_1), \ldots, f_n(\mathbf{u}_n)$ are multiplicatively dependent. We also prove, under certain conditions, a finiteness result for $\mathbf{u} \in \mathbb{Z}^m$ with relatively prime entries such that $f_1(\mathbf{u}),\ldots,f_n(\mathbf{u})$ are multiplicatively dependent.

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arXiv2024

On multiplicative dependence between elements of polynomial orbits

We classify the pairs of polynomials $f,g \in \mathbb{C}[X]$ having orbits satisfying infinitely many multiplicative dependence relations, extending a result of Ghioca, Tucker and Zieve. Moreover, we show that given $f_1,\ldots, f_n$ from a certain class of polynomials with integer coefficients, the vectors of indices $(m_1,\ldots,m_n)$ such that $f_1^{m_1}(0),\ldots,f_n^{m_n}(0)$ are multiplictively dependent are sparse. We also classify the pairs $f,g \in \mathbb{Q}[X]$ such that there are infinitely many $(x,y) \in \mathbb{Z}^2$ satisfying $f(x)^k=g(y)^\ell$ for some (possibly varying) non-zero integers $k,\ell$.

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arXiv2024

$S$-integral preperiodic points for monomial semigroups over number fields

We consider semigroup dynamical systems defined by several monnomials over a number field $K$. We prove a finiteness result for preperiodic points of such systems which are $S$-integral with respect to a non-preperiodic point $β$, which is uniform as $β$ varies over number fields of bounded degree. This generalises results of Baker, Ih and Rumely, which were made uniform by Yap, and verifies a special case of a natural generalisation of a conjecture of Ih.

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arXiv2022

Effective bounds on $S$-integral preperiodic points for polynomials

Given a polynomial $f$ defined over a number field $K$, we make effective certain special cases of a conjecture of S. Ih, on the finiteness of $f$-preperiodic points which are $S$-integral with respect to a fixed non-preperiodic point $α$. As an application, we obtain bounds on the number of $S$-units in the doubly indexed sequence $\{ f^n(α) - f^m(α) \}_{n > m \geq 0}$. In the case of a unicritical polynomial $f_c(z)=z^2+c$, with $α$ fixed to be the critical point 0, for parameters $c$ outside a small region, we give an explicit bound which depends only on the number of places of bad reduction for $f_c$. As part of the proof, we obtain novel lower bounds for the $v$-adically smallest preperiodic point of $f_c$ for each place $v$ of $K$.

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arXiv2019

On algebraic integers of bounded house and preperiodicity in polynomial semigroup dynamics

We consider semigroup dynamical systems defined by several polynomials over a number field $\mathbb{K}$, and the orbit (tree) they generate at a given point. We obtain finiteness results for the set of preperiodic points of such systems that fall in the cyclotomic closure of $\mathbb{K}$. More generally, we consider the finiteness of initial points in the cyclotomic closure for which the orbit contains an algebraic integer of bounded house. This work extends previous results for classical obits generated by one polynomial over $\mathbb{K}$ obtained initially by Dvornicich and Zannier (for preperiodic points), and then by Chen and Ostafe (for roots of unity and elements of bounded house in orbits).

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arXiv2018

On multiplicative independence of rational function iterates

We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a generalisation of Gao's method for constructing elements in the finite field $\mathbb{F}_{q^n}$ whose orders are larger than any polynomial in $n$ when $n$ becomes large. Additionally, we discuss the finiteness of polynomials which translate a given finite set of polynomials to become multiplicatively dependent.

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