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

Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that {\em counts} the number of irreducible factors of $f\bmod p$. We…

We study rational self-maps of $\mathbb{P}^{1}$ whose critical points all have finite forward orbit. Thurston's rigidity theorem states that outside a single well-understood family, there are finitely many such maps over $\mathbb{C}$ of…

Let $1<g_1<\ldots<g_{\varphi(p-1)}<p-1$ be the ordered primitive roots modulo~$p$. We study the pseudorandomness of the binary sequence $(s_n)$ defined by $s_n\equiv g_{n+1}+g_{n+2}\bmod 2$, $n=0,1,\ldots$. In particular, we study the…

We give a method of constructing polynomials of arbitrarily large degree irreducible over a global field F but reducible modulo every prime of F. The method consists of finding quadratic f in F[x] whose iterates have the desired property,…

Let $R$ be a real closed field. We consider basic semi-algebraic sets defined by $n$-variate equations/inequalities of $s$ symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by $2d < n$. Such a…

In his 1934 paper, G.\ Birkhoff poses the problem of classifying pairs $(G,U)$ where $G$ is an abelian group and $U\subset G$ a subgroup, up to automorphisms of $G$. In general, Birkhoff's Problem is not considered feasible. In this note,…

We present computational algorithms to work with points on the modular curve associated to the normaliser of a non-split Cartan group of prime level $p$. Rather than working with explicit equations, we represent these points using the…

The factor complexity ${\mathcal C}_{\mathbf u}$ of a sequence ${\mathbf u} = u_0u_1u_2 \cdots$ over a finite alphabet counts the number of factors of length $n$ occurring in $\mathbf u$, i.e., ${\mathcal C}_{\mathbf u}(n) = \#{\mathcal…

Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM $N=4$ theory and are known by previous work to be associated to positroids. In this paper we study the structure of the associated positroids, as well as…

Let $q = p^s$ be a power of a prime number $p$ and let $\mathbb{F}_q$ be the finite field with $q$ elements. In this paper we obtain the explicit factorization of the cyclotomic polynomial $\Phi_{2^nr}$ over $\mathbb{F}_q$ where both $r…

The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…

We address the primary decomposition of the knot concordance group in terms of the solvable filtration and higher-order von Neumann $\rho$-invariants by Cochran, Orr, and Teichner. We show that for a nonnegative integer n, if the connected…

For a graph $G$ on $[n]$, the $k$-cut complex $\Delta_k(G)$ has facets $[n]\setminus T$, where $T$ ranges over the disconnected $k$-vertex induced subgraphs of $G$. Bayer, Denker, Jeli\'c Milutinovi\'c, Sundaram, and Xue proved that the…

Let $ (\rho, V) $ be an irreducible representation of the symmetric group $ S_n$ (or the alternating group $ A_n$), and let $ g $ be a permutation on $n$ letters with each of its cycle lengths divides the length of its largest cycle. We…

Let $K$ be the function field of a smooth, irreducible curve defined over $\overline{\mathbb{Q}}$. Let $f\in K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r \ge 1,$ is a power of the prime number $p$, and let $\beta\in \overline{K}$.…

Morton and Vivaldi defined the polynomials whose roots are parabolic parameters for a one-parameter family of polynomial maps. We call these polynomials delta factors. They conjectured that delta factors are irreducible for the family…

Let g be a simple simply laced Lie algebra. In this paper two families of varieties associated to the Dynkin graph of g are described: ``tensor product'' and ``multiplicity'' varieties. These varieties are closely related to Nakajima's…

Let $G$ be a Cameron--Walker graph on $n$ vertices and $J_G$ the binomial edge ideal of $G$. Let $S$ denote the polynomial ring in $2n$ variables over a field. It is shown that the following conditions are equivalent: (i) $S/J_G$ is…

This note provides the complete factorization of the Misiurewicz-Thurston polynomial $q_{\ell,n}=p_{\ell+n}(z) - p_\ell(z)$ over $\mathbb{C}$, which plays a central role in the study of the Mandelbrot set, where \[ p_0(z) = 0, \qquad…

We first show the existence and nature of convergence to a limiting set of roots for polynomials in a three-term recurrence of the form $p_{n+1}(z) = Q_k(z)p_{n}(z)+ \gamma p_{n-1}(z)$ as $n$ $\rightarrow$ $\infty$, where the coefficient…