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The Lax conjecture is true
Author(s):
A.
S.
Lewis;
P.
A.
Parrilo;
M.
V.
Ramana
Journal:
Proc. Amer. Math. Soc.
133
(2005),
2495-2499.
MSC (2000):
Primary 15A45, 90C25, 52A41
Posted:
March 31, 2005
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Abstract:
In 1958 Lax conjectured that hyperbolic polynomials in three variables are determinants of linear combinations of three symmetric matrices. This conjecture is equivalent to a recent observation of Helton and Vinnikov.
References:
-
- 1.
- H.H. Bauschke, O. Güler, A.S. Lewis, and H.S. Sendov.
Hyperbolic polynomials and convex analysis.
Canadian Journal of Mathematics, 53:470-488, 2001. MR 1827817 (2002c:90099) - 2.
- C.B. Chua.
Relating homogeneous cones and positive definite cones via T-algebras.
SIAM Journal on Optimization, 14:500-506, 2003. MR 2048159 - 3.
- L. Faybusovich.
On Nesterov's approach to semi-infinite programming.
Acta Applicandae Mathematicae, 74:195-215, 2002. MR 1935854 (2003i:90094) - 4.
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Linear hyperbolic differential equations with constant coefficients.
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An inequality for hyperbolic polynomials.
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Hyperbolic polynomials and interior point methods for convex programming.
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Linear matrix inequality representation of sets.
Technical report, Mathematics Department, UCSD, 2002. - 8.
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Differential equations, difference equations and matrix theory.
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Self-adjoint determinantal representations of real plane curves.
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Additional Information:
A.
S.
Lewis
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Address at time of publication:
School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853
Email:
aslewis@sfu.ca, aslewis@orie.cornell.edu
P.
A.
Parrilo
Affiliation:
Automatic Control Laboratory, Swiss Federal Institute of Technology, CH-8092 Zürich, Switzerland
Email:
parrilo@control.ee.ethz.ch
M.
V.
Ramana
Affiliation:
Corporate Research and Development, United Airlines Inc., Elk Grove Village, Illinois 60007
Email:
motakuri_ramana@yahoo.com
DOI:
10.1090/S0002-9939-05-07752-X
PII:
S 0002-9939(05)07752-X
Keywords:
Hyperbolic polynomial,
Lax conjecture,
hyperbolicity cone,
semidefinite representable
Received by editor(s):
April 2, 2003
Posted:
March 31, 2005
Additional Notes:
The research of the first author was supported by NSERC
Communicated by:
Jonathan M. Borwein
Copyright of article:
Copyright
2005,
American Mathematical Society
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