arXiv subjects

Matt Hohertz

Publications and source records attributed to Matt Hohertz.

4 recordsLinked to original sources

arXiv2022

Bounds for zeros of Collatz polynomials, with necessary and sufficient strictness conditions

In a previous paper, we introduced the Collatz polynomials $P_N(z)$, whose coefficients are the terms of the Collatz sequence of the positive integer $N$. Our work in this paper expands on our previous results, using the Eneström-Kakeya Theorem to tighten our old bounds of the roots of $P_N(z)$ and giving precise conditions under which these new bounds are sharp. In particular, we confirm an experimental result that zeros on the circle $\{z\in\mathbb{C}: |z| = 2\}$ are rare: the set of $N$ such that $P_N(z)$ has a root of modulus 2 is sparse in the natural numbers. We close with some questions for further study.

math.CV↗

arXiv2021

An extension of the Geometric Modulus Principle to holomorphic and harmonic functions

Kalantari's Geometric Modulus Principle describes the local behavior of the modulus of a polynomial. Specifically, if $p(z) = a_0 + \sum_{j=k}^n a_j\left(z-z_0\right)^j,\;a_0a_ka_n \neq 0$, then the complex plane near $z = z_0$ comprises $2k$ sectors of angle $\fracπ{k}$, alternating between arguments of ascent (angles $θ$ where $|p(z_0 + te^{iθ})| > |p(z_0)|$ for small $t$) and arguments of descent (where the opposite inequality holds). In this paper, we generalize the Geometric Modulus Principle to holomorphic and harmonic functions. As in Kalantari's original paper, we use these extensions to give succinct, elegant new proofs of some classical theorems from analysis.

math.CV↗

arXiv2020

A constructive proof of the convergence of Kalantari's bound on polynomial zeros

In his 2006 paper, Jin proves that Kalantari's bounds on polynomial zeros, indexed by $m \leq 2$ and called $L_m$ and $U_m$ respectively, become sharp as $m\rightarrow\infty$. That is, given a degree $n$ polynomial $p(z)$ not vanishing at the origin and an error tolerance $ε> 0$, Jin proves that there exists an $m$ such that $\frac{L_m}{ρ_{min}} > 1-ε$, where $ρ_{min} := \min_{ρ:p(ρ) = 0} \left|ρ\right|$. In this paper we derive a formula that yields such an $m$, thereby constructively proving Jin's theorem. In fact, we prove the stronger theorem that this convergence is uniform in a sense, its rate depending only on $n$ and a few other parameters. We also give experimental results that suggest an optimal m of (asymptotically) $O\left(\frac{1}{ε^d}\right)$ for some $d \ll 2$. A proof of these results would show that Jin's method runs in $O\left(\frac{n}{ε^d}\right)$ time, making it efficient for isolating polynomial zeros of high degree.

math.CV↗

arXiv2020

Collatz polynomials: an introduction with bounds on their zeros

The Collatz Conjecture (also known as the 3x+1 Problem) proposes that the following algorithm will, after a certain number of iterations, always yield the number 1: given a natural number, multiply by three and add one if the number is odd, halve the resulting number, then repeat. In this article, for each $N$ for which the Collatz Conjecture holds we define the $N^{th}$ Collatz polynomial to be the monic polynomial with constant term $N$ and $k^{th}$ term (for $k > 1$) the $k^{th}$ iterate of $N$ under the Collatz function. In particular, we bound the moduli of the roots of these polynomials, prove theorems on when they have rational integer roots, and suggest further applications and avenues of research.

math.NT↗