Related papers: Capture components of cubic Siegel polynomials

A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the integer lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry.…

Let $f(x)=x^n+ax^2+bx+c \in \Z[x]$ be an irreducible polynomial with $b^2=4ac$ and let $K=\Q(\theta)$ be an algebraic number field defined by a complex root $\theta$ of $f(x)$. Let $\Z_K$ deonote the ring of algebraic integers of $K$. The…

Let $G$ be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of G, we determine its Langlands parameter (equivalently, its Jordan blocks…

In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of the curves X^2+3Y^2 = 4DZ^3 and…

In complex dynamics, the boundaries of higher dimensional hyperbolic components in holomorphic families of polynomials or rational maps are mysterious objects, whose topological and analytic properties are fundamental problems. In this…

The aim of the present short note is to answer the open questions posted by Hern\'andez, Martin, and Rodrigues in {\rm \cite{p1,p2}}. The obtained results give the complete classification of irreducible components in the varieties of Jordan…

We generalize the construction from arXiv:2102.09329 of theta series for quadratic forms of signature $(n-1,1)$ with homogeneous and spherical polynomials. Namely, we allow that the parameters $c_1,c_2$, which define the theta series and…

We consider cubic forms $\phi_{a,b}(x,y,z) = ax^3 + by^3 - z^3$ with coefficients $a,b \in \mathbb{Z}$. We give an asymptotic formula for how many of these forms are locally soluble everywhere, i.e. we give an asymptotic formula for the…

Given a separable nonconstant polynomial $f(x)$ with integer coefficients, we consider the set $S$ consisting of the squarefree parts of all the rational values of $f(x)$, and study its behavior modulo primes. Fixing a prime $p$, we…

We constructed Yoccoz puzzle for cosine functions $f(z)=ae^z+be^{-z}$ with bounded post-critical set, and proved that a Fatou component is a Jordan domains if it is bounded and is not eventually a Siegal disk. We proved that $f$ is…

For any polynomial f with complex coefficients we find a remarkable subset of poles of the motivic zeta function. It is combinatorially determined by any log resolution and it admits an intrinsic interpretation in terms of contact loci of…

Let $\mathcal Q_D$ be the set of moduli points on Siegel's modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order $\mathcal O$ in an indefinite quaternion algebra of…

We consider three kinds of quotients of the curve complex which are obtained by coning off uniformly quasi-convex subspaces: symmetric curve sets, non-maximal train track sets, and compression body disc sets. We show that the actions of the…

We prove that the moduli space ${\mathcal A}_{g,\Gamma_0(p)}\otimes \bar {\mathbb F}_p$ of principally polarized abelian varieties of dimension $g$ with a $\Gamma_0(p)$-level structure in characteristic $p$ has $2^g$ irreducible…

Let $F$ be a field of characteristic zero and $W$ be an associative affine $F$-algebra satisfying a polynomial identity (PI). The codimension sequence associated to $W$, $c_n(W)$, is known to be of the form $\Theta (c n^t d^n)$, where $d$…

We formulate a detailed conjectural Eichler-Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes $\mathrm{GU}(2,1,\mathbb{Q}(\sqrt{-3}))$. The formula is based…

Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to…

We determine the support of the irreducible spherical representation (i.e., the irreducible quotient of the polynomial representation) of the rational Cherednik algebra of a finite Coxeter group for any value of the parameter c. In…

In this paper, we study rigidity of polynomials of arbitrary degree in the presence of neutral dynamics. Specifically, we focus on {non-renormalizable} (in the sense of Douady and Hubbard) complex polynomials of degree $d \geqslant 2$ that…

We prove new bounds for the Fourier coefficients of Jacobi forms using a method of Iwaniec. In view of the Fourier-Jacobi expansion of degree two Siegel modular forms, we can use these to obtain strong bounds on fundamental Fourier…