5,438 research outputs found
The differential Hilbert function of a differential rational mapping can be computed in polynomial time
International audienceWe present a probabilistic seminumerical algorithm that computes the differential Hilbert function associated to a differential rational mapping. This algorithm explicitly determines the set of variables and derivatives which can be arbitrarily fixed in order to locally invert the differential mapping under consideration. The arithmetic complexity of this algorithm is polynomial in the input size
Modular embeddings of Teichmueller curves
Fuchsian groups with a modular embedding have the richest arithmetic
properties among non-arithmetic Fuchsian groups. But they are very rare, all
known examples being related either to triangle groups or to Teichmueller
curves.
In Part I of this paper we study the arithmetic properties of the modular
embedding and develop from scratch a theory of twisted modular forms for
Fuchsian groups with a modular embedding, proving dimension formulas,
coefficient growth estimates and differential equations.
In Part II we provide a modular proof for an Apery-like integrality statement
for solutions of Picard-Fuchs equations. We illustrate the theory on a worked
example, giving explicit Fourier expansions of twisted modular forms and the
equation of a Teichmueller curve in a Hilbert modular surface.
In Part III we show that genus two Teichmueller curves are cut out in Hilbert
modular surfaces by a product of theta derivatives. We rederive most of the
known properties of those Teichmueller curves from this viewpoint, without
using the theory of flat surfaces. As a consequence we give the modular
embeddings for all genus two Teichmueller curves and prove that the Fourier
developments of their twisted modular forms are algebraic up to one
transcendental scaling constant. Moreover, we prove that Bainbridge's
compactification of Hilbert modular surfaces is toroidal. The strategy to
compactify can be expressed using continued fractions and resembles
Hirzebruch's in form, but every detail is different.Comment: revision including the referee's comments, to appear in Compositio
Mat
Cauchy Type Integrals of Algebraic Functions
We consider Cauchy type integrals with an algebraic function. The main goal is to give
constructive (at least, in principle) conditions for to be an algebraic
function, a rational function, and ultimately an identical zero near infinity.
This is done by relating the Monodromy group of the algebraic function , the
geometry of the integration curve , and the analytic properties of the
Cauchy type integrals. The motivation for the study of these conditions is
provided by the fact that certain Cauchy type integrals of algebraic functions
appear in the infinitesimal versions of two classical open questions in
Analytic Theory of Differential Equations: the Poincar\'e Center-Focus problem
and the second part of the Hilbert 16-th problem.Comment: 58 pages, 19 figure
Six results on Painleve VI
After recalling some of the geometry of the sixth Painleve equation, we will
describe how the Okamoto symmetries arise naturally from symmetries of
Schlesinger's equations and summarise the classification of the Platonic
Painleve six solutions. A key observation is that Painleve VI governs the
isomonodromic deformations of certain Fuchsian systems on rank \emph{three}
bundles.Comment: 16 pages, written for Angers 2004 International Conference on
Asymptotic Theories and Painleve Equations (updated, added Remark 7 giving
simple formulae for isomonodromic families of rank three systems in terms of
PVI solutions and as an example the full family of "Klein connections" are
written down
Dimensional analysis using toric ideals: Primitive invariants
© 2014 Atherton et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.Classical dimensional analysis in its original form starts by expressing the units for derived quantities, such as force, in terms of power products of basic units M, L, T etc. This suggests the use of toric ideal theory from algebraic geometry. Within this the Graver basis provides a unique primitive basis in a well-defined sense, which typically has more terms than the standard Buckingham approach. Some textbook examples are revisited and the full set of primitive invariants found. First, a worked example based on convection is introduced to recall the Buckingham method, but using computer algebra to obtain an integer K matrix from the initial integer A matrix holding the exponents for the derived quantities. The K matrix defines the dimensionless variables. But, rather than this integer linear algebra approach it is shown how, by staying with the power product representation, the full set of invariants (dimensionless groups) is obtained directly from the toric ideal defined by A. One candidate for the set of invariants is a simple basis of the toric ideal. This, although larger than the rank of K, is typically not unique. However, the alternative Graver basis is unique and defines a maximal set of invariants, which are primitive in a simple sense. In addition to the running example four examples are taken from: a windmill, convection, electrodynamics and the hydrogen atom. The method reveals some named invariants. A selection of computer algebra packages is used to show the considerable ease with which both a simple basis and a Graver basis can be found.The third author received funding from Leverhulme Trust Emeritus Fellowship (1-SST-U445) and United Kingdom EPSRC grant: MUCM EP/D049993/1
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