Michael F. Singer

Galois Theory of Linear Differential Equations

Linear differential equations form the central topic of this volume, Galois theory being the unifying theme. A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilberts 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used. This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

Liouvillian first integrals of differential equations

Liouvillian functions are functions that are built up from rational functions using exponentiation, integration, and algebraic functions. We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liouvillian function of several variables vanishing on the curve defined by this solution, then the system has a liouvillian first integral, that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set. We can refine this result in special cases to show that the first integral must be of a very special form

Galois Theory of Difference Equations

Picard-Vessiot rings.- Algorithms for difference equations.- The inverse problem for difference equations.- The ring S of sequences.- An excursion in positive characteristic.- Difference modules over .- Classification and canonical forms.- Semi-regular difference equations.- Mild difference equations.- Examples of equations and galois groups.- Wild difference equations.- q-difference equations.

Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields

The authors consider the problem of reconstructing (i.e., interpolating) a t-sparse multivariate polynomial given a black box which will produce the value of the polynomial for any value of the arguments. It is shown that, if the polynomial has coefficients in a finite field

Galois groups of second and third order linear differential equations

GF[q]

Liouvillian Solutions of n-th Order Homogeneous Linear Differential Equations

and the black box can evaluate the polynomial in the field

Testing reducibility of linear differential operators: A group theoretic perspective

GF[q^{\ulcorner 2\log_{q}(nt)+3 \urcorner}]

Liouvillian and algebraic solutions of second and third order linear differential equations

, where n is the number of variables, then there is an algorithm to interpolate the polynomial in

Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients

O(\log^3 (nt))

Solving Difference Equations in Finite Terms

boolean parallel time and