Michael F. Singer

Universität Heidelberg IWR INF 368 (2008)

Abstract. McGrail [3] has shown the existence of a model completion for the universal theory of fields on which a finite number of commuting derivations act and, independently, Yaffe [7] has shown...

Zoé Chatzidakis, Charlotte Hardouin, Michael F. Singer

We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable...

1 Introduction to the Galois Theory of Linear Differential Equations (2008)

1 Introduction to the Galois Theory of Linear Differential Equations (2008)

Linear Differential Operators for Polynomial Equations (2008)

Olivier Cormier, Michael F. Singer, Barry M. Trager, Barry M, Felix Ulmer, ...

Given a squarefree polynomial P k 0 [x, y], k 0 a number field, we construct a linear differential operator that allows one to calculate the genus of the complex curve defined by P = 0 (when P is...

Differential Galois Theory of Linear Difference Equations (2008)

Hardouin, Charlotte, Singer, Michael F.

We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hoelder's Theorem...

Introduction to the Galois Theory of Linear Differential Equations (2007)

maths.univ-rennes1.fr (2007)

This is an expanded version of the 10 lectures given as the 2006 London Mathematical Society Invited Lecture Series at the Heriot-Watt University 31 July - 4 August 2006.

3 1 (2007)

Olivier Cormier, Michael F. Singer, Felix Ulmer, ...

In this paper we show how to compute the geometric Galois group G Q(x) of a polynomial f 2 Q(x)[Y] by considering the associated linear differential equation L f (Y) = 0 (and constructions of it)...

Olivier Cormier, Campus De Beaulieu, Michael F. Singer, Felix Ulmer

maths.univ-rennes1.fr In this paper we show how to compute the Galois group G of a polynomial f 2 Q(x)[Y] by factoring the associated linear differential equation Lf (Y) = 0 (and constructions of it)...

Linear Dierential Operators for Polynomial Equations (2007)

Olivier Cormier, Felix Ulmer, Michael F. Singer, Barry M. Trager

Given a squarefree polynomial P 2 k 0 [x; y], k 0 a number eld, we construct a linear dierential operator that allows one to calculate the genus of the complex curve dened by P = 0 (when P is...

and (2007)

Claude Mitschi, Michael F. Singer

The usual Galois theory of polynomial equations allows one to associate a group to a polynomial in such a way that the algebraic properties of the roots of the polynomial are reflected in properties...

third (2007)

algorithm for liouvillian solutions of

third (2007)

On the Definitions of Difference Galois Groups (2007)

algorithm for liouvillian solutions of

Chatzidakis, Zoé, Hardouin, Charlotte, Singer, Michael F.

We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable...

A Recursive Method for Determining the One-Dimensional Submodules of Laurent-Ore Modules (2006)

Li, Ziming, Singer, Michael F., Wu, Min, Zheng, Dabin

We present a method for determining the one-dimensional submodules of a Laurent-Ore module. The method is based on a correspondence between hyperexponential solutions of associated systems and...

Model Theory of Partial Differential Fields: From Commuting to Noncommuting Derivations (2006)

Measuring Presence in Virtual Environments (2006)

McGrail has shown the existence of a model completion for the universal theory of fields on which a finite number of commuting derivations act and, independently, Yaffe has shown the existence of a...

Witmer, Bob G., Singer, Michael F.

A primary argument for the efficacy of Virtual Environments (VE) applications is that the user is 'present' in the simulated environment. Presence is defined as the subjective experience of being in...

Galois theory of parameterized differential equations and linear differential algebraic groups (2006)

Phyllis J. Cassidy, Michael F. Singer

linear differential algebraic groups

Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups (2005)

Cassidy, Phyllis J., Singer, Michael F.

We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of...

On the constructive inverse problem in differential Galois theory (2005)

William J. Cook, Claude Mitschi, Michael F. Singer

We give sufficient conditions for a differential equation to have a given semisimple group as its Galois group. For any group G with G 0 = G1 ·... · Gr where each Gi is a simple group of type Aℓ,...

On the constructive inverse problem in differential Galois theory (2005)

William J. Cook, Claude Mitschi, Michael F. Singer

We give sufficient conditions for a differential equation to have a given semisimple group as its Galois group. For any group G with G 0 = G 1 · ·· · ·G r, where each G i is a simple group of...

On the Constructive Inverse Problem in Differential Galois Theory (2004)

Cook, William J., Mitschi, Claude, Singer, Michael F.

We give sufficient conditions for a linear differential equation to have a given semisimple group as its Galois group. For any linear algebraic group G given as a semidirect product of a finite...

Solvable-by-finite groups as differential Galois groups (2002)

Solvable-by-finite groups as differential Galois groups (2002)

Solvable-by-finite groups as differential Galois groups (2002)

We prove the inverse problem of differential Galois theory over the differential field k=C(x), where C is an algebraic closed field of characteristic zero, for linear algebraic groups G over CC with...

Linear differential operators for polynomial equations (2002)

We prove the inverse problem of differential Galois theory over the differential field k=C(x), where C is an algebraic closed field of characteristic zero, for linear algebraic groups G over CC with...

Olivier Cormier, Michael F. Singer, Barry M. Trager, Felix Ulmer

Given a squarefree polynomial P ∈ k0[x, y], k0 a number field, we construct a linear differential operator that allows one to calculate the genus of the complex curve defined by P = 0 (when P is...

Computing Galois Groups of Completely Reducible Differential Equations (1998)

Elie Compoint, Michael F. Singer

this paper, we will show that for differential equations whose Galois group is reductive, one can effectively present the corresponding Picard-Vessiot extension and from this presentation compute the...

Liouvillian and Algebraic Solutions of Second and Third Order Linear Differential Equations (1998)

Galois Groups of Second and Third Order Linear Differential Equations (1997)

this paper we show that the index of a 1-reducible subgroup of the differential Galois

Linear Differential Equations and Products of Linear Forms (1997)

this paper we show how factorization properties of these symmetric powers can be used to determine structural properties of the galois groups of second and third order linear differential equation....

Direct and Inverse Problems in Differential Galois Theory (1997)

We show that liouvillian solutions of an n-th order linear differential equation L(y) = 0 are related to semi-invariant forms of the differential Galois group of L(y) = 0 which factor into linear...

On Magid's approach to the inverse problem in differential Galois theory. (1996)

. This paper surveys recent work on the problems of calculating Galois groups of differentialequations and of constructing differentialequations with given groups as their Galois groups. 1....

Kovacic, Jerry, Mitschi, Claude, Singer, Michael F.

We present counterexamples to Theorem 7.3 and Theorem 7.13 in {em Lectures in Differential Galois Theory} by A. Magid, University Lecture Series, Vol. 7, AMS 1994.

On Magid's approach to the inverse problem in differential Galois theory. (1996)

Kovacic, Jerry, Mitschi, Claude, Singer, Michael F.

We present counterexamples to Theorem 7.3 and Theorem 7.13 in {em Lectures in Differential Galois Theory} by A. Magid, University Lecture Series, Vol. 7, AMS 1994.

Testing reducibility of linear differential operators: a group theoretic perspective (1996)

Connected Linear Groups as Differential Galois Groups over $C(x)$. (1995)

Let k be an ordinary differential field of characteristic zero and let D = k[D] be the ring of linear differential operators over k, that is, the noncommutative polynomial ring in the

Connected Linear Groups as Differential Galois Groups over $C(x)$. (1995)

We generalize results of Kovacic to solve the inverse problem in differential Galois theory for connected linear groups, over $C(x)$ where $C$ is an arbitrary algebraically closed field $C$ of...

On Ramis's solution of the local inverse problem of differential Galois theory. (1994)

We generalize results of Kovacic to solve the inverse problem in differential Galois theory for connected linear groups, over $C(x)$ where $C$ is an arbitrary algebraically closed field $C$ of...

On Ramis's solution of the local inverse problem of differential Galois theory. (1994)

Recently, J.P. Ramis gave necessary and sufficient conditions for a linear algebraic group to be the Galois group of a Picard-Vessiot extension of the field ${\bf C}\{x\}[x^{-1}]$ of germs of...

Moduli of linear differential equations on the Riemann sphere with fixed Galois groups. (1993)

Recently, J.P. Ramis gave necessary and sufficient conditions for a linear algebraic group to be the Galois group of a Picard-Vessiot extension of the field ${\bf C}\{x\}[x^{-1}]$ of germs of...

Necessary Conditions for Liouvillian Solutions of (Third Order) Linear Differential Equations (1993)

Liouvillian and Algebraic Solutions of Second and Third Order Linear Differential Equations (1993)

In this paper we show how group theoretic information can be used to derive a set of necessary conditions on the coefficients of L(y) for L(y) = 0 to have a liouvillian solution. The method is used...

Moduli of linear differential equations on the Riemann sphere with fixed Galois groups (1993)

this paper we show that the index of a 1-reducible subgroup of the differential Galois group of an ordinary homogeneous linear differential equation L(y) = 0 yields the best possible bound for the...

Computational Complexity of Sparse Rational Interpolation (1991)

For fixed m and n, we consider the vector space of linear differential equations of order n whose coefficients are polynomials of degree at most m. We show that for G in a large class of linear...

Dima Grigoriev, Marek Karpinski, Michael F. Singer

We analyze the computational complexity of sparse rational interpolation, and give the first genuine time (arithmetic complexity does not depend on the size of the coefficients) algorithm for this...

AND (1987)

Lee A. Rubel, Michael F. Singer

By definition, an autonomous function is a differentially algebraic function Jon iw (or on C), every translate J, of which satisfies every algebraic differential equation that J satisfies. We find...

Functions satisfying elementary relations (1977)

Elementary solutions of differential equations. (1975)

1

Functions satisfying elementary relations. (1974)

Functions satisfying elementary relations / (1974)

Thesis (Ph. D. in Mathematics)--Univ. of California, Berkeley, June 1974.