47 research outputs found
A Difference Version of Nori's Theorem
We consider (Frobenius) difference equations over (F_q(s,t), phi) where phi
fixes t and acts on F_q(s) as the Frobenius endomorphism. We prove that every
semisimple, simply-connected linear algebraic group G defined over F_q can be
realized as a difference Galois group over F_{q^i}(s,t) for some i in N. The
proof uses upper and lower bounds on the Galois group scheme of a Frobenius
difference equation that are developed in this paper. The result can be seen as
a difference analogue of Nori's Theorem which states that G(F_q) occurs as
(finite) Galois group over F_q(s).Comment: 29 page
Parameterized Picard-Vessiot extensions and Atiyah extensions
Generalizing Atiyah extensions, we introduce and study differential abelian
tensor categories over differential rings. By a differential ring, we mean a
commutative ring with an action of a Lie ring by derivations. In particular,
these derivations act on a differential category. A differential Tannakian
theory is developed. The main application is to the Galois theory of linear
differential equations with parameters. Namely, we show the existence of a
parameterized Picard-Vessiot extension and, therefore, the Galois
correspondence for many differential fields with, possibly, non-differentially
closed fields of constants, that is, fields of functions of parameters. Other
applications include a substantially simplified test for a system of linear
differential equations with parameters to be isomonodromic, which will appear
in a separate paper. This application is based on differential categories
developed in the present paper, and not just differential algebraic groups and
their representations.Comment: 90 pages, minor correction
The Fuchsian differential equation of the square lattice Ising model susceptibility
Using an expansion method in the variables that appear in the
-dimensional integrals representing the -particle contribution to the
Ising square lattice model susceptibility , we generate a long series of
coefficients for the 3-particle contribution , using a
polynomial time algorithm. We give the Fuchsian differential equation of order
seven for that reproduces all the terms of our long series. An
analysis of the properties of this Fuchsian differential equation is performed.Comment: 15 pages, no figures, submitted to J. Phys.
Groupes de Galois differentiels et G-fonctions
SIGLECNRS T Bordereau / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc
Divergent series, summability and resurgence I: monodromy and resurgence
Providing an elementary introduction to analytic continuation and monodromy, the first part of this volume applies these notions to the local and global study of complex linear differential equations, their formal solutions at singular points, their monodromy and their differential Galois groups. The Riemann-Hilbert problem is discussed from Bolibrukh’s point of view. The second part expounds 1-summability and Ecalle’s theory of resurgence under fairly general conditions. It contains numerous examples and presents an analysis of the singularities in the Borel plane via “alien calculus”, which provides a full description of the Stokes phenomenon for linear or non-linear differential or difference equations. The first of a series of three, entitled Divergent Series, Summability and Resurgence, this volume is aimed at graduate students, mathematicians and theoretical physicists interested in geometric, algebraic or local analytic properties of dynamical systems. It includes useful exercises with solutions. The prerequisites are a working knowledge of elementary complex analysis and differential algebra