Related papers: Transfinite diameter and the resultant

In \cite{Bedford}, the dynamics of a particular polynomial diffeomorphism of $\mathbb{C}^N$, called a polynomial shift-like map of type $\nu$, has been studied as a higher dimensional analog of H\'enon maps. In this note, we prove that the…

We prove a strong general-purpose bound for the diameter of a finite group depending only on the diameters of its composition factors and the maximal exponent of a normal abelian section. There are a number of notable applications: (1) if…

We call shifted power a polynomial of the form $(x-a)^e$. The main goal of this paper is to obtain broadly applicable criteria ensuring that the elements of a finite family $F$ of shifted powers are linearly independent or, failing that, to…

Let $g(z)=\int_0^zp(t)\exp(q(t))\,dt+c$ where $p,q$ are polynomials and $c\in\mathbb{C}$, and let $f$ be the function from Newton's method for $g$. We show that under suitable assumptions the Julia set of $f$ has Lebesgue measure zero.…

We obtain the convergence speed for Fekete points on uniformly polynomially cuspidal compact sets introduced by Pawlucki and Ple\'sniak. This is done by showing that these sets are $(\mathscr{C}^{\alpha}, \mathscr{C}^{\alpha'})$-regular in…

Expressions are given for the exponential of a hermitian matrix, A. Replacing A by iA these are explicit formulas for the Fourier transform of exp(iA). They extend to any size matrix the previous results for the 2 X 2, 3 X 3, and 4 X 4…

We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial $f$ have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge\exp(|z|^\alpha)$ for…

If M is a Drinfeld module over a local function field L, we may view M as a dynamical system, and consider its filled Julia set J. If J^0 is the connected component of the identity, relative to the Berkovich topology, we give a…

In this paper we deal with a finite abelian group $G$ and the abstract Fourier transform ${\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$. Then, we consider $\tilde{j}\circ {\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$ where $\tilde…

In this paper we give a unified proof of the fact that the Julia set of Newton's method applied to a holomorphic function of the complex plane (a polynomial of degree large than $1$ or an entire transcendental function) is connected. The…

Let U(L) be the enveloping algebra of a finite dimensional Lie algebra L over a field k of characteristic zero, Z(U(L)) its center and Sz(U(L)) its semicenter. A sufficient condition is given in order for Sz(U(L)) to be a polynomial algebra…

Discrete trigonometric transformations, such as the discrete Fourier and cosine/sine transforms, are important in a variety of applications due to their useful properties. For example, one well-known property is the convolution theorem for…

In this article, we present a distance estimation formula that can be used to ray trace 3D slices of the filled-in Julia sets and the Multibrot sets generated by the tricomplex polynomials of the form $\eta^p+c$ where $p$ is any integer…

Let $R$ be a commutative Noetherian local ring. We prove a variety of new formulae for modules of finite quasi-projective or finite quasi-injective dimension. These include the Derived Depth Formula, itself an extension of Auslander famous…

Let $f$ be Fatou's function, that is, $f(z)= z+1+e^{-z}$. We prove that the escaping set of $f$ has the structure of a `spider's web' and we show that this result implies that the non-escaping endpoints of the Julia set of $f$ together with…

Following partially a suggestion by Pyber, we prove that the diameter of a product of non-abelian finite simple groups is bounded linearly by the maximum diameter of its factors. For completeness, we include the case of abelian factors and…

Helfgott and Seress have proved the existence of a quasipolynomial upper bound on the diameter of $\mathrm{Alt}(n)$. In this paper, we walk partway towards removing the dependence on CFSG from that result, by using the algorithm solving the…

We study the Fourier transform of the absolute value of a polynomial on a finite-dimensional vector space over a local field of characteristic 0. We prove that this transform is smooth on an open dense set. We prove this result for the…

Let $E \subseteq \mathbb{R}^n$ be a union of line segments and $F \subseteq \mathbb{R}^n$ the set obtained from $E$ by extending each line segment in $E$ to a full line. Keleti's line segment extension conjecture posits that the Hausdorff…

We prove strong and explicit diameter bounds for finite simple Lie algebras, which parallel Babai's conjecture for finite simple groups. Specifically, we show that any nonabelian finite simple Lie algebra $\mathfrak{g}$ over $\mathbf{F}_p$…