Enumerative real algebraic geometry
Frank Sottile · DIMACS series in discrete mathematics and theoretical computer science · 2003
Contents 1. Introduction 1 2. Sparse Polynomial Systems 2 3. Enumerative Real Algebraic Geometry 10 4. Schubert Calculus 18 5. The Conjecture of Shapiro and Shapiro 27 6. Lower Bounds in the Schubert calculus 33 Acknowledgements 37 References 37 1. Introduction Consider the following question. Question 1.1. Find a priori information about the number of real solutions to a structured system of real polynomial equations 0 = f 1 = f 2 = = f N ; where each f i 2 R[x 1 ; : : : ; x n ] : (1.1) Two well-dened classes of structured polynomial systems have been studied from this point of view|sparse systems, where the structure is encoded by the monomials in the polynomials f i |and geometric systems, where the structure co