Related papers: Necklaces, permutations, and periodic critical orb…

It is proved that for every complex quadratic polynomial $f$ with Cremer's fixed point $z_0$ (or periodic orbit) for every $\delta>0$, there is at most one periodic orbit of minimal period $n$ for all $n$ large enough, entirely in the disc…

It is known that, when $n$ is even, the number of permutations of $\{1,2,\dots,n\}$ all of whose cycles have odd length equals the number of those all of whose cycles have even length. Adin, Heged\H{u}s and Roichman recently found a…

For $n \geq 3$ and $s \leq 92$, it is proved in \cite{ShSi} that, except for finitely many pairs $(n, s), G_1(x) = G_1(x, n, s) $ is either irreducible or linear factor times an irreducible polynomial. If $s \leq 30$, we determine here…

Let $g_t$ be a loop in the space of monic complex polynomials in one variable of fixed degree $n$. If the roots of $g_t$ are distinct for all $t$, they form a braid $B_1$ on $n$ strands. Likewise, if the critical points of $g_t$ are…

Let {T_n} be the bipolar filtration of the smooth concordance group of topologically slice knots, which was introduced by Cochran, Harvey, and Horn. It is known that for each n not equal to 1 the quotient group T_n/T_{n+1} has infinite rank…

Consider the moduli space, $\mathcal{M}_{d}$, of degree $d \geq 2$ polynomials over $\mathbb{C}$, with a marked critical point. Given $k \geq 0,\; p$ an odd prime, we show that the set $\Sigma_{k,1,p}$ of conjugacy classes of degree $p$…

The multiplier $\lambda_n$ of a periodic orbit of period $n$ can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials $p_c(z)=z^2+c$. We provide a numerical algorithm for computing critical…

Let $G$ be one of the Artin groups of finite type ${\mathbf B}_n={\mathbf C}_n$, and affine type $\tilde{\mathbf A}_{n-1}$ and $\tilde{\mathbf C}_{n-1}$. In this paper, we show that if $\alpha$ and $\beta$ are elements of $G$ such that…

Let $n$ and $k$ be positive integers, and $f_n(k)$ (resp. $g_n(k)$) be the number of unital subrings (resp. unital irreducible subrings) of $\mathbb{Z}^n$ of index $k$. The numbers $f_n(k)$ are coefficients of certain zeta functions of…

There has been interest during the last decade in properties of the sequence {gcd(a^n-1,b^n-1)}, n=1,2,3,..., where a,b are fixed (multiplicatively independent) elements in either the rational integers, the polynomials in one variable over…

We study the number of irreducible factors (over $\mathbb{Q}$) of the $n$th iterate of a polynomial of the form $f_r(x) = x^2 + r$ for rational $r$. When the number of such factors is bounded independent of $n$, we call $f_r(x)$…

Let T be a finite subset of the complex unit circle S^1, and define f: S^1 -> S^1 by f(z) = z^d. Let CH(T) denote the convex hull of T. If card(T) = N > 2, then CH(T) defines a polygon with N sides. The N-gon CH(T) is called a…

Let $p_c(\mathbb{Q}_n)$ and $p_c(\mathbb{Z}^n)$ denote the critical values for nearest-neighbour bond percolation on the $n$-cube $\mathbb{Q}_n = \{0,1\}^n$ and on $\Z^n$, respectively. Let $\Omega = n$ for $\mathbb{G} = \mathbb{Q}_n$ and…

Let $\Bbb F_q$ be a finite field with $q$ elements. Let $n$ be a positive integer with radical $rad(n)$, namely, the product of distinct prime divisors of $n$. If the order of $q$ modulo $rad(n)$ is either 1 or a prime, then the irreducible…

Let $G$ and $N$ be finite groups of order $2n$ where $n$ is odd. We say the pair $(G,N)$ is Hopf-Galois realizable if $G$ is a regular subgroup of $\h(N)=N\rtimes\au(N)$. In this article we give necessary conditions on $G$ (similarly $N$)…

Fix a finite alphabet. A necklace is a circular word. For positive integers $n$ and~$k$, a necklace is $(n,k)$-perfect if all words of length $n$ occur $k$ times but at positions with different congruence modulo $k$, for any convention of…

We introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting…

We show that detecting real roots for honestly n-variate (n+2)-nomials (with integer exponents and coefficients) can be done in time polynomial in the sparse encoding for any fixed n. The best previous complexity bounds were exponential in…

The article presents results on the well-known problem concerning the structure of integer polynomials $p_n(z; x, y)$, which define multiplication laws in $n$-valued groups $\mathbb{G}_n$ over the field of complex numbers $\mathbb{C}$. We…

The space of monic centered cubic polynomials with marked critical points is isomorphic to C^2. For each n>0, the locus Sn formed by all polynomials with a specified critical point periodic of exact period n forms an affine algebraic set.…