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Mark van Hoeij

Publications and source records attributed to Mark van Hoeij.

34 records · Page 2Linked to original sources
arXiv2016

A Modular Algorithm for Computing Polynomial GCDs over Number Fields presented with Multiple Extensions

We consider the problem of computing the monic gcd of two polynomials over a number field L = Q(alpha_1,...,alpha_n). Langemyr and McCallum have already shown how Brown's modular GCD algorithm for polynomials over Q can be modified to work for Q(alpha) and subsequently, Langemyr extended the algorithm to L[x]. Encarnacion also showed how to use rational number to make the algorithm for Q(alpha) output sensitive, that is, the number of primes used depends on the size of the integers in the gcd and not on bounds based on the input polynomials. Our first contribution is an extension of Encarnacion's modular GCD algorithm to the case n>1, which, like Encarnacion's algorithm, is is output sensitive. Our second contribution is a proof that it is not necessary to test if p divides the discriminant. This simplifies the algorithm; it is correct without this test. Our third contribution is a modification to the algorithm to treat the case of reducible extensions. Such cases arise when solving systems of polynomial equations. Our fourth contribution is an implementation of the modular GCD algorithm in Maple and in Magma. Both implementations use a recursive dense polynomial data structure for representing polynomials over number fields with multiple field extensions. Our fifth contribution is a primitive fraction-free algorithm. This is the best non-modular approach. We present timing comparisons of the Maple and Magma implementations demonstrating various optimizations and comparing them with the monic Euclidan algorithm and our primitive fraction-free algorithm.

cs.SC↗
arXiv2014

Belyi functions for hyperbolic hypergeometric-to-Heun transformations

A complete classification of Belyi functions for transforming certain hypergeometric equations to Heun equations is given. The considered hypergeometric equations have the local exponent differences 1/k,1/l,1/m that satisfy k,l,m in N and the hyperbolic condition 1/k+1/l+1/m<1. There are 366 Galois orbits of Belyi functions giving the considered (non-parametric) hypergeometric-to-Heun pull-back transformations. Their maximal degree is 60, which is well beyond reach of standard computational methods. To obtain these Belyi functions, we developed two efficient algorithms that exploit the implied pull-back transformations.

math.AG↗
arXiv2014

Gonality of the modular curve X1(N)

In this paper we compute the gonality over Q of the modular curve X1(N) for all N <= 40 and give upper bounds for each N <= 250. This allows us to determine all N for which X1(N) has infinitely points of degree <= 8. We conjecture that the modular units of Q(X1(N)) are freely generated by f_2,...,f_{[N/2]+1} where f_k is obtained from the equation for X1(k).

math.NT↗
arXiv2013

Algorithms and differential relations for Belyi functions

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 on implied transformations of the hypergeometric differential equation to Fuchsian equations, preferably with few singular points. This gives interesting differential relations between polynomial components of a Belyi function.

math.AG↗
arXiv2012

On stringy invariants of GUT vacua

We investigate aspects of certain stringy invariants of singular elliptic fibrations which arise in engineering Grand Unified Theories in F-theory. In particular, we exploit the small resolutions of the total space of these fibrations provided recently in the physics literature to compute `stringy characteristic classes', and find that numerical invariants obtained by integrating such characteristic classes are predetermined by the topology of the base of the elliptic fibration. Moreover, we derive a simple (dimension independent) formula for pushing forward powers of the exceptional divisor of a blowup, which one may use to reduce any integral (in the sense of Chow cohomology) on a small resolution of a singular elliptic fibration to an integral on the base. We conclude with a speculatory note on the cohomology of small resolutions of GUT vacua, where we conjecture that certain simple formulas for their Hodge numbers may be given solely in terms of the first Chern class and Hodge numbers of the base.}

math.AG↗
arXiv2012

On Hirzebruch invariants of elliptic fibrations

We compute all Hirzebruch invariants $χ_q$ for $D_5$, $E_6$, $E_7$ and $E_8$ elliptic fibrations of every dimension. A single generating series $χ(t,y)$ is produced for each family of fibrations such that the coefficient of $t^{k}y^{q}$ encodes $χ_q$ over a base of dimension $k$, solely in terms of invariants of the base of the fibration.

math.AG↗
arXiv2011

Explicit formula for the generating series of diagonal 3D rook paths

Let $a_n$ denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an $n \times n \times n$ three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven \emph{discovery and proof} of the fact that the generating series $G(x)= \sum_{n \geq 0} a_n x^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function: \[ G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.\]

cs.SC↗
arXiv2010

Isomorphisms of Algebraic Number Fields

Let $\mathbb{Q}(α)$ and $\mathbb{Q}(β)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $\mathbb{Q}(β) \rightarrow \mathbb{Q}(α)$. The algorithm is particularly efficient if the number of isomorphisms is one.

cs.SC↗
arXiv2004

Solving conics over Q(t1,..,tk)

Let K = Q(t1,..,tk) and a,b,c in K. We give a simple algorithm to find, if it exists, X,Y,Z in K, not all zero, for which aX^2 + bY^2 + cZ^2 = 0.

math.NT↗
arXiv2002

Computing Riemann Theta Functions

The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-) periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation are given. First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision. This formula is used to construct a uniform approximation formula, again with arbitrary precision.

nlin.SI↗