Related papers: Bounding the degrees of a minimal $\mu$-basis for …

We prove a lower bound on the canonical height associated to polynomials over number fields evaluated at points with infinite forward orbit. The lower bound depends only on the degree of the polynomial, the degree of the number field, and…

In this paper we discuss the relationship between the moving planes of a rational parametric surface and the singular points on it. Firstly, the intersection multiplicity of several planar curves is introduced. Then we derive an equivalent…

For a given branched covering between closed connected surfaces, there are several easy relations one can establish between the Euler characteristics of the surfaces, their orientability, the total degree, and the local degrees at the…

The degree of a map between orientable manifolds is a fundamental concept in topology, offering deep insights into the structure of the manifolds and the nature of the corresponding maps. This concept has been extensively studied,…

The goal of this paper is to accurately describe the maximal zero-free region of the independence polynomial for graphs of bounded degree, for large degree bounds. In previous work with de Boer, Guerini and Regts it was demonstrated that…

In this paper we study sets of points in the plane with rational distances from r prescribed points P_1, ...,P_r. A crucial case arises for r = 3, where we provide simple necessary and sufficient conditions for the density of this set in…

In this paper, we study triply periodic surfaces with minimal surface area under a constraint in the volume fraction of the regions (phases) that the surface separates. Using a variational level set method formulation, we present a…

Let $\mathbb{F}_q$ be a finite field with $q=p^n$ elements. In this paper, we study the number of $\mathbb{F}_q$-rational points on the affine hypersurface $\mathcal X$ given by $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$, where…

We establish the existence and uniqueness of rational conformal maps of minimal degree $n+1$ for opening up $n$ arcs. In earlier results, the degree was exponential in $n$. We also discuss two related problems. (a) We establish existence of…

We study the dynamics of polynomial-like mappings in several variables. A special case of our results is the following theorem. Let f be a proper holomorphic map from an open set U onto a Stein manifold V, $U\subset\subset V$. Assume f is…

Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we…

We investigate the structure of polynomials of degree four in many variables over a fixed prime field $\mathbb{F}=\mathbb{F}_{p}$. In 2007, Green and Tao proved that if a polynomial $f:\mathbb{F}^{n}\rightarrow\mathbb{F}$ is poorly…

Given a polarized variety $(X,L)$, we construct and study projections of low degree $ X\dashrightarrow \mathbb{P}(H^0(L^\vee)) \dashrightarrow \mathbb P ^n $ using the associated kernel bundles. As an application, we can show that the…

We obtain explicit factorizations of reduced period polynomials of degree $2^m$, $m\ge 4$, for finite fields of characteristic $p\equiv 3$ or $5\pmod{8}$. This extends the results of G. Myerson, who considered the cases $m=1$ and $m=2$, and…

For each integer $g\geq 1$ we use variational methods to construct in the unit $3$-ball $B$ a free boundary minimal surface $\Sigma_g$ of symmetry group $\mathbb{D}_{g+1}$. For $g$ large, $\Sigma_g$ has three boundary components and genus…

We consider positive solutions to parametrized systems of generalized polynomial equations (with real exponents and positive parameters). By a fundamental result obtained in parallel work, polynomial systems are determined by geometric…

We discuss methods to construct a polynomial parametrization of some interesting knotted surfaces (knotted spheres, knotted tori and knotted planes) and provide examples.

A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which…

The goal of this paper is to construct a compactification of the moduli space of degree $d \ge 5$ surfaces in $\mathbb{P}^3$, i.e. a parameter space whose interior points correspond to (equivalence classes of) smooth surfaces in…

In 2018, by Ramsey and Hoffman theory, Koolen, Yang, and Yang presented a structural result on graphs with smallest eigenvalue at least $-3$ and large minimum degree. In this study, we depart from the conventional use of Ramsey theory and…