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On computing Belyi maps

J. Sijsling, J. Voight (2014)

Publications mathématiques de Besançon

We survey methods to compute three-point branched covers of the projective line, also known as Belyĭ maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and p -adic methods. Along the way, we pose several questions and provide numerous examples.

On computing subfields. A detailed description of the algorithm

Jürgen Klüners (1998)

Journal de théorie des nombres de Bordeaux

Let ( α ) be an algebraic number field given by the minimal polynomial f of α . We want to determine all subfields ( β ) ( α ) of given degree. It is convenient to describe each subfield by a pair ( g , h ) [ t ] × [ t ] such that g is the minimal polynomial of β = h ( α ) . There is a bijection between the block systems of the Galois group of f and the subfields of ( α ) . These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding...

On reduced Arakelov divisors of real quadratic fields

Ha Thanh Nguyen Tran (2016)

Acta Arithmetica

We generalize the concept of reduced Arakelov divisors and define C-reduced divisors for a given number C ≥ 1. These C-reduced divisors have remarkable properties, similar to the properties of reduced ones. We describe an algorithm to test whether an Arakelov divisor of a real quadratic field F is C-reduced in time polynomial in l o g | Δ F | with Δ F the discriminant of F. Moreover, we give an example of a cubic field for which our algorithm does not work.

On the class numbers of real cyclotomic fields of conductor pq

Eleni Agathocleous (2014)

Acta Arithmetica

The class numbers h⁺ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows, and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed, which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime...

On the computation of Hermite-Humbert constants for real quadratic number fields

Michael E. Pohst, Marcus Wagner (2005)

Journal de Théorie des Nombres de Bordeaux

We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields. With these algorithms we are able to compute extreme Humbert forms for the number fields ( 13 ) and ( 17 ) . Finally we compute the Hermite-Humbert constant for the number field ( 13 ) .

On the computation of quadratic 2 -class groups

Wieb Bosma, Peter Stevenhagen (1996)

Journal de théorie des nombres de Bordeaux

We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer D 0 , 1 mod 4 and the factorization of D , computes the structure of the 2 -Sylow subgroup of the class group of the quadratic order of discriminant D in random polynomial time in log D .

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