Related papers: Hyperbolic components and cubic polynomials

We show that a one-ended simply connected at infinity hyperbolic group $G$ with enough codimension-1 surface subgroups has $\partial G \cong \mathbb{S}^2$. Combined with a result of Markovic, our result gives a new characterization of…

We consider Jordan curves of the form $\gamma=\cup_{j=1}^n \gamma_j$ on the Riemann sphere for which each $\gamma_j$ is a hyperbolic geodesic in $(\widehat{\mathbb C} \smallsetminus \gamma)\cup \gamma_j$. These Jordan curves are…

For a semialgebraic set K in R^n, let P_d(K) be the cone of polynomials in R^n of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary. When K=R^n and d is even, we show that its boundary lies on the…

Let $J_r$ denote an $r\times r$ matrix over a finite field $F$ with minimal and characteristic polynomials $(t-1)^r$. Suppose $r\leq s$. It is not hard to show that the Jordan canonical form of $J_r\otimes J_s$ is similar to…

Supersymmetric curves are the analogue of Riemann surfaces in super geometry. We establish some foundational results about complex Deligne-Mumford superstacks, and we then prove that the moduli superstack of supersymmetric curves is a…

Semilinear elliptic equations of the form $-\Delta u =\lambda|u|^{p-2}u- |u|^{q-2}u$ in bounded and unbounded domains are considered. In the plane of exponents $p\times q$, the so-called critical exponents curve is introduced which…

Consider the parameter space $\mathcal{P}_{\lambda}\subset \mathbb{C}^{2}$ of complex H\'enon maps $$ H_{c,a}(x,y)=(x^{2}+c+ay,ax),\ \ a\neq 0 $$ which have a semi-parabolic fixed point with one eigenvalue $\lambda=e^{2\pi i p/q}$. We give…

Let $1\leq p \leq +\infty$. We show that the positive part of the closed unit ball of a non-commmutative $L^p$-space, as a metric space, is a complete Jordan $^*$-invariant for the underlying von Neumann algebra.

Let $E$ be an elliptic curve defined over a number field $K$. We say that a prime number $p$ is exceptional for $(E,K)$ if $E$ admits a $p$-isogeny defined over $K$. The so-called exceptional set of all such prime numbers is finite if and…

We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given…

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 $p$…

To every hyperelliptic curve one can assign the periods of the integrals over the holomorphic and the meromorphic differentials. By comparing two representations of the so-called projective connection it is possible to reexpress the latter…

We use a global version of Heath-Brown's $p-$adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most $B$ on non-singular cubic curves defined over $\mathbb{Q}$. The bounds are…

We prove that cubic polynomial maps with a fixed Siegel disk and a critical orbit eventually landing inside that Siegel disk lie in the support of the bifurcation measure. This answers a question of Dujardin in positive. Our result implies…

For a given elliptic curve $E$ over a finite local ring, we denote by $E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ can be described solely in terms of its $x$-coordinate $P_x$, which can be therefore used to…

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials.…

The tricorn, the connectedness locus of the anti-holomorphic quadratic family, is known to be non-locally connected. The boundary of every hyperbolic component of odd period contains arcs that are inaccessible from the complement of the…

We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthemore, any hyperbolic metric on the torus with cone singularities of positive curvature can be…

We show that every polynomial of degree $d \geq 2$ in the connectedness locus with an attracting cycle which attracts at least two critical points and no indifferent cycles is not combinatorially rigid. In particular, we prove that a…

Given a polynomial $p$, the degree of its Chebyshev's method $C_p$ is determined. If $p$ is cubic then the degree of $C_p$ is found to be $4,6$ or $7$ and we investigate the dynamics of $C_p$ in these cases. If a cubic polynomial $p$ is…