EDITORS’ PREFACE xi (5) Joos Heintz, Bart Kuijpers and Andr´ Rojas Paredes, “On the intrinsic complexity of elimination problems in effective algebraic geome- try”. The representation of polynomials by arithmetic circuits evaluating them is an alternative data structure which allowed considerable progress in polynomial equation solving in the last fifteen years. The authors present in this contribution a circuit based computation model which captures the core of all known symbolic elimination algorithms that avoid unneces- sary branchings in effective algebraic geometry and show the intrinsically exponential complexity character of elimination in this complexity model. (6) Gregorio Malajovich, “Newton iteration, conditioning and zero count- ing”. This contribution deals with the problem of counting the number of real solutions of a system of multivariate polynomial equations with real coef- ficients. You can also find in this contribution a crash-course in Newton iteration. We will state and analyze a Newton iteration based ‘inclusion- exclusion’ algorithm to count (and find) roots of real polynomials. In recent months, two members of our scientific community left us: our col- league Mario Wschebor and our beloved friend Jean-Pierre Dedieu. Jean-Pierre was invited to the Summer School and his talk was scheduled as the closing talk of the School. Unfortunately, a long illness prevented him from being with us at the School and, sadly, he left us on 15 June 2012. Let this volume serve as a remembrance of both of them. The editors wish to thank the RSME for giving us the opportunity to organize this event. It is also a pleasure to thank the patronage of the UIMP. Their help in the organization and the experience in Las Caballerizas del Palacio de la Mag- dalena are not to be easily forgotten. Our deepest gratitude goes to the speakers, who did an excellent job, and also to the students, whose interest and dedication created a great atmosphere. We finally wish to thank the authors for their excellent contributions to this volume. Jos´ e Luis Monta˜ na & Luis M. Pardo
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