• 1 Introduction
  • 2 Semi-algebraic Exchange Economies
  • 2.1 Mathematical Preliminaries I:Polynomial Rings and Semi-Algebraic Sets and Functions
  • 2.2 Semi-algebraic Economies
  • 3 The Main Result
  • 3.1 From Competitive Equilibrium to Polynomial Equations
  • 3.2 Mathematical Preliminaries III: GrÄobner Bases and the Shape Lemma
  • 3.3 Proof of Theorem 1 and Discussion
  • 3.4 Arbitrary Parameters
  • 3.5 Practical Implications: Equilibrium Multiplicity
  • 4 Applications
  • 4.1 Quadratic Utility
  • 4.2 CES Utility
  • 5 Incomplete Financial Markets
  • 5.1 Models with Incomplete Financial Markets
  • 5.2 Example
  • 6 Conclusion
  • References