- 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
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