Real Solving for Positive Dimensional Systems

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 Gröbner basis methods for finding 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 finding all roots of a single polynomial in a single unknown. The software package Singular computes the equivalent simple system. If all coefficients in the original equilibrium equations are rational numbers or parameters then the Gröbner basis computations of Singular are exact. This fact implies that the Gröbner 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 Gröbner basis methods can be easily applied to gain interesting insights into many modern economic models. * We thank seminar participants at the 2007 and 2008 Institute for Computational Economics at the University of Chicago for comments. We thank Gerhard Pfister for help with Singular and are grateful to Gerhard Pfister and Bernd Sturmfels for patiently answering our questions on computational algebraic geometry.

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“…positive dimensional). Aubry et al (2002) develop an algorithm to find representative solutions of the positive dimensional system.…”

Section: Detecting Multiple Solutionsmentioning

confidence: 99%

Tackling Multiplicity of Equilibria with Gröbner Bases

Kübler

Schmedders

2010

Operations Research

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“…For univariate f (n = 1) Sturm sequences [9] yield an efficient algorithm for deciding semidefiniteness. The bivariate case n = 2 can be solved by Seidenberg's [26] algorithm (see also [9] and [11]), which is generalized to arbitrarily many variables via Lagrangian multipliers in [1,25] or used in nonstandard decision methods [29]. Alternatively, one can use Artin's theorem of sumof-squares and semidefinite programming (see, e.g., [12,14]).…”

Section: Motivationmentioning

confidence: 99%

Computing the radius of positive semidefiniteness of a multivariate real polynomial via a dual of Seidenberg's method

Hutton

Kaltofen

Zhi

2010

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

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We give a stability criterion for real polynomial inequalities with floating point or inexact scalars by estimating from below or computing the radius of semidefiniteness. That radius is the maximum deformation of the polynomial coefficient vector measured in a weighted Euclidean vector norm within which the inequality remains true. A large radius means that the inequalities may be considered numerically valid.The radius of positive (or negative) semidefiniteness is the distance to the nearest polynomial with a real root, which has been thoroughly studied before. We solve this problem by parameterized Lagrangian multipliers and Karush-Kuhn-Tucker conditions. Our algorithms can compute the radius for several simultaneous inequalities including possibly additional linear coefficient constraints. Our distance measure is the weighted Euclidean coefficient norm, but we also discuss several formulas for the weighted infinity and 1-norms.The computation of the nearest polynomial with a real root can be interpreted as a dual of Seidenberg's method that decides if a real hypersurface contains a real point. Sums-of-squares rational lower bound certificates for the radius of semidefiniteness provide a new approach to solving Seidenberg's problem, especially when the coefficients are numeric. They also offer a surprising alternative sum-of-squares proof for those polynomials that themselves cannot be represented by a polynomial *

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“…Then, if (x (1) ,μ (1) ), (x (2) ,μ (2) ) are two different solutions of (11), x (1) =x (2) . By taking into account every B ⊂ {2, .…”

Section: Definition 19mentioning

confidence: 99%

On Sign Conditions Over Real Multivariate Polynomials

Jeronimo

Perrucci

Sabia

2009

Discrete Comput Geom

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We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that intersect transversally. This enables us to show a probabilistic procedure to list all feasible sign conditions over the polynomials. In addition, we extend these results to the case of closed sign conditions over an arbitrary family of real multivariate polynomials. The complexity bounds for these procedures improve the known ones.

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Real Solving for Positive Dimensional Systems

Cited by 61 publications

References 17 publications

Tackling Multiplicity of Equilibria with Gröbner Bases

Tackling Multiplicity of Equilibria with Gröbner Bases

Computing the radius of positive semidefiniteness of a multivariate real polynomial via a dual of Seidenberg's method

On Sign Conditions Over Real Multivariate Polynomials

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