# Tackling multiplicity of equilibria with Gröbner bases

Kübler, Felix; Schmedders, Karl (2010). Tackling multiplicity of equilibria with Gröbner bases. Operations Research, 58(4):1037-1050.

## Abstract

Multiplicity of equilibria is a prevalent problem in many economic models. Often equilibria are characterized as solutions to a system of polynomial equations. This paper gives an introduction to the application of GrÄobner basis methods for ¯nding all solutions of a polynomial system. The Shape Lemma, a key result from algebraic geometry, states under mild assumptions that a given equilibrium system has the same solution set as a much simpler triangular system. Essentially the computation of all solutions then reduces to ¯nding all roots of a single polynomial in a single unknown. The software package Singular computes the equivalent simple system. If all coeficients in the original equilibrium equations are rational numbers or parameters then the GrÄobner basis computations of Singular are exact. This fact implies that the GrÄobner basis methods cannot only be used for a numerical approximation of equilibria but in fact may allow the proof of theoretical results for the underlying economic model.
Three economic applications illustrate that without much prior knowledge of algebraic geometry GrÄobner basis methods can be easily applied to gain interesting insights into many modern economic models.

## Abstract

Multiplicity of equilibria is a prevalent problem in many economic models. Often equilibria are characterized as solutions to a system of polynomial equations. This paper gives an introduction to the application of GrÄobner basis methods for ¯nding all solutions of a polynomial system. The Shape Lemma, a key result from algebraic geometry, states under mild assumptions that a given equilibrium system has the same solution set as a much simpler triangular system. Essentially the computation of all solutions then reduces to ¯nding all roots of a single polynomial in a single unknown. The software package Singular computes the equivalent simple system. If all coeficients in the original equilibrium equations are rational numbers or parameters then the GrÄobner basis computations of Singular are exact. This fact implies that the GrÄobner basis methods cannot only be used for a numerical approximation of equilibria but in fact may allow the proof of theoretical results for the underlying economic model.
Three economic applications illustrate that without much prior knowledge of algebraic geometry GrÄobner basis methods can be easily applied to gain interesting insights into many modern economic models.

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