xii Preface theories. Our text includes complete proofs, both of the results on algebraic geometry and algebraic groups which are needed in Picard-Vessiot theory and of the results on Picard-Vessiot theory itself. We have given several courses on Differential Galois Theory in Barcelona and Krak´ ow. As a result, we published our previous book “Introduction to Differential Galois Theory” [C-H1]. Although published by a university publishing house, it has made some impact and has been useful to graduate students as well as to theoretical physicists working on dynamical systems. Our present book is also aimed at graduate students in mathematics or physics and at researchers in these fields looking for an introduction to the subject. We think it is suitable for a graduate course of one or two semesters, depending on students’ backgrounds in algebraic geometry and algebraic groups. Interested students can work out the exercises, some of which give an insight into topics beyond the ones treated in this book. The prerequisites for this book are undergraduate courses in commutative algebra and complex analysis. We would like to thank our colleagues Jos´ e Mar´ ıa Giral, Andrzej No- wicki, and Henryk ˙ Zoadek who carefully read parts of this book and made valuable comments, as well as Jakub Byszewski and S lawomir Cynk for interesting discussions on its content. We are also grateful to the anonymous referees for their corrections and suggestions which led to improvements in the text. Our thanks also go to Dr. Ina Mette for persuading us to expand our previous book to create the present one and for her interest in this project. Finally our book owes much to Jerry Kovacic. We will always be thankful to him for many interesting discussions and will remember him as a brilliant mathematician and an open and friendly person. Both authors acknowledge support by Spanish Grants MTM2006-04895 and MTM2009-07024, Polish Grant N20103831/3261 and European Network MRTN-CT-2006-035495. Barcelona and Krak´ ow, October 2010 Teresa Crespo and Zbigniew Hajto
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