Related papers: The Belyi degree is computable
Belyi's theorem asserts that a smooth projective curve $X$ defined over a number field can be realized as a cover of the projective line unramified outside three points. In this short paper we investigate the bejaviour of the minimal degree…
In this short note we introduce the Belyi degree of a number field K, which is the smallest degree of a dessin d'enfant having K as field of moduli. After the description of some general properties (for example, the fact that there exist…
We give an elementary, self-contained and quick proof of Belyi's theorem. As a by-product of our proof we obtain an explicit bound for the degree of the defining number field of a Belyi surface.
Belyi's Theorem states that a Riemann surface, X, as an algebraic curve is defined over an algebraic closure of the rationals if and only if there exists a holomorphic function taking X to the Riemann sphere with at most three critical…
A tool package for computing genus 0 Belyi functions is presented, including simplification routines, computation of moduli fields, decompositions, dessins d'enfant. The main algorithm for computing the Belyi functions themselves is based…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
We use a numerical method to compute a database of three-point branched covers of the complex projective line of small degree. We report on some interesting features of this data set, including issues of descent.
The purpose of the present paper is to give an effective version of the noncritical $p$-tame Belyi theorem. That is to say, we compute explicitly an upper bound of the minimal degree of tamely ramified Belyi maps in positive characteristic…
We survey methods to compute three-point branched covers of the projective line, also known as Belyi maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic…
We study the Betti tables of reducible algebraic curves with a focus on connected line arrangements and provide a general formula for computing the quadratic strand of the Betti table for line arrangements that satisfy certain hypotheses.…
Let $\pi$ be a factor code from a one dimensional shift of finite type $X$ onto an irreducible sofic shift $Y$. If $\pi$ is finite-to-one then the number of preimages of a typical point in $Y$ is an invariant called the degree of $\pi$. In…
Given smooth, projective, geometrically integral algebraic curves $X$ and $Y$ defined over a number field $K$, assuming that there is a non-constant $K$-morphism $\varphi \colon X \to Y$, we give an upper bound on the minimum of the degrees…
In this elementary note we prove that a polynomial with rational coefficients divides the derivative of some polynomial which splits in $\Q$ if and only if all of its irrational roots are real and simple. This provides an answer to a…
When $p$ is a computable real so that $p \geq 1$, the isometry degree of a computable copy $\mathcal{B}$ of $\ell^p$ is defined to be the least powerful Turing degree that computes a linear isometry of $\ell^p$ onto $\mathcal{B}$. We show…
Motivated by a demand for explicit genus 1 Belyi maps from theoretical physics, we give an efficient method of explicitly computing genus one Belyi maps by (1) composing covering maps from elliptic curves to the Riemann sphere with simpler…
An almost Belyi covering is an algebraic covering of the projective line, such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line. In general, there are 1-dimensional families of…
The degree of a curve $C$ in a polarized abelian variety $(X,\lambda)$ is the integer $d=C\cdot\lambda$. When $C$ generates $X$, we find a lower bound on $d$ which depends on $n$ and the degree of the polarization $\lambda$. The smallest…
We develop in this article an algorithm that, given a projective curve $C$, computes a \textit{gonal map}, that is, a finite morphism from $C$ to the projective line of minimal degree. Our method is based on the computation of scrollar…
We present an algorithmic way of exactly computing Belyi functions for hypermaps and triangulations in genus 0 or 1, and the associated dessins, based on a numerical iterative approach initialized from a circle packing combined with…
In this article we consider rational functions on algebraic curves, which have one zero and one pole (and call pair of such function and curve Abel pair). We investigate moduli spaces of such functions on curves of genus one; the number of…