arXiv subjects

Taboka Chalebgwa

Publications and source records attributed to Taboka Chalebgwa.

3 recordsLinked to original sources

arXiv2022

SIN, COS, EXP and LOG of Liouville numbers

For any Liouville number $α$, all of the following are transcendental numbers: $\textrm{e}^α$, $\log_\textrm{e}α$, $\sin α$, $\cosα$, $\tanα$, $\sinhα$, $\coshα$, $\tanhα$, $\arcsinα$ and the inverse functions evaluated at $α$ of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if "Liouville number" is replaced by "$U$-number", where $U$ is one of Mahler's classes of transcendental numbers.

math.NT↗

arXiv2020

On algebraic values of Weierstrass $σ$-functions

Suppose that $Ω$ is a lattice in the complex plane and let $σ$ be the corresponding Weierstrass $σ$-function. Assume that the point $τ$ associated to $Ω$ in the standard fundamental domain has imaginary part at most 1.9. Assuming that $Ω$ has algebraic invariants $g_2,g_3$ we show that a bound of the form $c d^m (\log H)^n$ holds for the number of algebraic points of height at most $H$ and degree at most $d$ lying on the graph of $σ$. To prove this we apply results by Masser and Besson. What is perhaps surprising is that we are able to establish such a bound for the whole graph, rather than some restriction. We prove a similar result when, instead of $g_2,g_3$, the lattice points are algebraic. For this we naturally exclude those $(z,σ(z))$ for which $z\inΩ$.

math.NT↗

arXiv2018

Sendov's Conjecture: A note on a paper of Dégot

Sendov's conjecture states that if all the zeroes of a complex polynomial $P(z)$ of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of $P(z)$. In a paper that appeared in 2014, Dégot proved that, for each $a\in (0,1)$, there exists an integer $N$ such that for any polynomial $P(z)$ with degree greater than $N$, if $P(a) = 0$ and all zeroes lie inside the unit disk, the disk $|z-a|\leq 1$ contains a critical point of $P(z)$. Based on this result, we derive an explicit formula $\mathcal{N}(a)$ for each $a \in (0,1)$ and, consequently obtain a uniform bound $N$ for all $a\in [α, β]$ where $0<α< β< 1$. This (partially) addresses the questions posed in Dégot's paper.

math.CV↗