Related papers: The arc length of a random lemniscate

Let $n \geq 1$, and let $p : {\bf C} \to {\bf C}$ be a monic polynomial of degree $n$. It was conjectured by Erd\H{o}s, Herzog, and Piranian that the maximal length of lemniscate $\{z \in {\bf C}: |p(z)| = 1\}$ is attained by the polynomial…

Let p(z) be a monic polynomial of degree n. Consider the lemniscate L={z:|p(z)|=1}. It has been conjectured that L has the largest length when p(z)=z^n-1. We show that the length of L attains a local maximum at this polynomial and prove the…

A lemniscate of a complex polynomial $Q_n$ of degree $n$ is a sublevel set of its modulus, i.e., of the form $\{z \in \mathbb{C}: |Q_n(z)| < t\}$ for some $t>0.$ In general, the number of connected components of this lemniscate can vary…

We show that for a monic polynomial p of degree d, the length of the level set {z: |p(z)|=1} is at most 9.2 d, which improves an earlier estimate due to P. Borwein. For d=2 we show that the extremal level set is the Bernoullis' Lemniscate.…

To each generic complex polynomial $p(z)$ there is associated a labeled binary tree (here referred to as a "lemniscate tree") that encodes the topological type of the graph of $|p(z)|$. The branching structure of the lemniscate tree is…

A classically studied geometric property associated to a complex polynomial $p$ is the inradius (the radius of the largest inscribed disk) of its (filled) lemniscate $\Lambda := \{z \in \mathbb{C}:|p(z)| < 1\}$. In this paper, we study the…

We prove that for every $0 < c < 4$ and every $N \in \mathbb{N}$ there exists a monic polynomial $p(z) = z^n + a_{n-1} z^{n-1} + \dots + a_0$ such that the set $\{z \in \mathbb{C} : |p(z)| \leq 1\}$ has at least $N$ connected components…

Let $K\subset\mathbb{C}$ be a compact set in the plane whose logarithmic capacity $c(K)$ is strictly positive. Let $\mathscr{P}_n(K)$ be the space of monic polynomials of degree $n,$ \emph{all} of whose zeros lie in $K.$ For $p\in…

Consider a sequence of random polynomials $P_n(z) = \prod_{k=1}^{n}(z - X_k)$, where $\{X_k\}_k$ are i.i.d. random variables distributed uniformly on the unit disc $\mathbb{D}$. Let $\Lambda_n = \{z \in \mathbb{C}: |P_n(z)| < 1\}$ be the…

We consider a family of heights defined by the $L_p$ norms of polynomials with respect to the equilibrium measure of a lemniscate for $0 \le p \le \infty$, where $p=0$ corresponds to the geometric mean (the generalized Mahler measure) and…

We prove that lemniscates (i.e., sets of the form $|P(z)|=1$ where $P$ is a complex polynomial) are irreducible real algebraic curves.

We investigate the geometry of a random rational lemniscate $\Gamma$, the level set $\{|r(z)|=1\}$ on the Riemann sphere of the modulus of a random rational function $r$. We assign a probability distribution to the space of rational…

Let $p_n$ denote the number of self-avoiding polygons of length $n$ on a regular three-dimensional lattice, and let $p_n(K)$ be the number which have knot type $K$. The probability that a random polygon of length $n$ has knot type $K$ is…

We answer a question of Erd\"os, Herzog, and Piranian on the minimal area of polynomial lemniscates when all the zeros of the polynomial are constrained to lie on a compact set K whose logarithmic capacity is strictly larger than 1.

For a non-constant complex rational function $P$, the lemniscate of $P$ is defined as the set of points $z\in \mathbb C$ such that $\vert P(z)\vert =1$. The lemniscate of $P$ coincides with the set of real points of the algebraic curve…

A rational lemniscate is a level set of $|r|$ where $r: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}$ is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This…

We discuss the probability of random knotting for a model of self-avoiding polygons whose segments are given by cylinders of unit length with radius $r$. We show numerically that the characteristic length of random knotting is roughly…

Erd\"os posed in 1940 the extremal problem of studying the minimal area of the lemniscate $\{|p(z)|<1\}$ of a monic polynomial $p$ of degree $n$ all of whose zeros are in the closed unit disc. In this article, we prove that there exist…

In this paper we sharpen significantly several known estimates on the maximal number of zeros of complex harmonic polynomials. We also study the relation between the curvature of critical lemniscates and its impact on geometry of caustics…

We give statistical definitions of the length, l, of a loose prime knot tied into a long, fluctuating ring macromolecule. Monte Carlo results for the equilibrium, good solvent regime show that < l > ~ N^t, where N is the ring length and t ~…