Displaying similar documents to “Computing fundamental domains for Fuchsian groups”

On computing quaternion quotient graphs for function fields

Gebhard Böckle, Ralf Butenuth (2012)

Journal de Théorie des Nombres de Bordeaux

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Let Λ be a maximal 𝔽 q [ T ] -order in a division quaternion algebra over 𝔽 q ( T ) which is split at the place ∞ . The present article gives an algorithm to compute a fundamental domain for the action of the group of units Λ * on the Bruhat-Tits tree 𝒯 associated to PGL 2 ( 𝔽 q ( ( 1 / T ) ) ) . This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group Λ * in terms of generators and relations. Moreover we determine an...

Solving conics over function fields

Mark van Hoeij, John Cremona (2006)

Journal de Théorie des Nombres de Bordeaux

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Let F be a field whose characteristic is not  2 and K = F ( t ) . We give a simple algorithm to find, given a , b , c ∈ K * , a nontrivial solution in  K (if it exists) to the equation a X 2 + b Y 2 + c Z 2 = 0 . The algorithm requires, in certain cases, the solution of a similar equation with coefficients in F ; hence we obtain a recursive algorithm for solving diagonal conics over ℚ ( t 1 , ⋯ , t n ) (using existing algorithms for such equations over  ℚ ) and over 𝔽 q ( t 1 , ⋯ , t n ) .

Factoring polynomials over global fields

Karim Belabas, Mark van Hoeij, Jürgen Klüners, Allan Steel (2009)

Journal de Théorie des Nombres de Bordeaux

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We prove that van Hoeij’s original algorithm to factor univariate polynomials over the rationals runs in polynomial time, as well as natural variants. In particular, our approach also yields polynomial time complexity results for bivariate polynomials over a finite field.

On the preconditioned biconjugate gradients for solving linear complex equations arising from finite elements

Michal Křížek, Jaroslav Mlýnek (1994)

Banach Center Publications

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The paper analyses the biconjugate gradient algorithm and its preconditioned version for solving large systems of linear algebraic equations with nonsingular sparse complex matrices. Special emphasis is laid on symmetric matrices arising from discretization of complex partial differential equations by the finite element method.