Contents

• Contents
• Introduction
• Three Classes of Polynomial Systems
• The Principles of Polynomial Homotopy Continuation Methods
• The Geometry of the Deformations
• Root Counts and Start Systems
  • Dense Polynomials modeled by Highest Degrees
  • Mixed Subdivisions of Newton Polytopes to compute Mixed Volumes
  • Sparse Polynomial Systems solved by Polyhedral Homotopies
  • Determinantal Polynomials arising in Enumerative Geometry
• Numerical Software for Solving Polynomial Systems
• The Database of Applications
• Closing Remarks and Open Problems
• Bibliography