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Multivariate stable polynomials: theory and applications
Author:
David G. Wagner
Journal:
Bull. Amer. Math. Soc. 48 (2011), 53-84
MSC (2010):
Primary 32A60; Secondary 05A20, 05B35, 15A45, 15A48, 60G55, 60K35
Posted:
October 4, 2010
MathSciNet review:
2738906
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Additional Information
Abstract: Univariate polynomials with only real roots, while special, do occur often enough that their properties can lead to interesting conclusions in diverse areas. Due mainly to the recent work of two young mathematicians, Julius Borcea and Petter Brändén, a very successful multivariate generalization of this method has been developed. The first part of this paper surveys some of the main results of this theory of multivariate stable polynomials--the most central of these results is the characterization of linear transformations preserving stability of polynomials. The second part presents various applications of this theory in complex analysis, matrix theory, probability and statistical mechanics, and combinatorics.
- 1.
Julius
Borcea and Petter
Brändén, Applications of stable polynomials to mixed
determinants: Johnson’s conjectures, unimodality, and symmetrized
Fischer products, Duke Math. J. 143 (2008),
no. 2, 205–223. MR 2420507
(2009b:15015), http://dx.doi.org/10.1215/00127094-2008-018
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Julius
Borcea and Petter
Brändén, Lee-Yang problems and the geometry of
multivariate polynomials, Lett. Math. Phys. 86
(2008), no. 1, 53–61. MR 2460727
(2010e:82038), http://dx.doi.org/10.1007/s11005-008-0271-6
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Julius
Borcea and Petter
Brändén, Pólya-Schur master theorems for
circular domains and their boundaries, Ann. of Math. (2)
170 (2009), no. 1, 465–492. MR 2521123
(2010g:30004), http://dx.doi.org/10.4007/annals.2009.170.465
- 4.
J. Borcea and P. Brändén, Multivariate Pólya-Schur classification problems in the Weyl algebra, Proc. London Math. Soc. 101 (2010), 73-104.
- 5.
Julius
Borcea and Petter
Brändén, The Lee-Yang and Pólya-Schur programs.
I. Linear operators preserving stability, Invent. Math.
177 (2009), no. 3, 541–569. MR 2534100
(2011g:47069), http://dx.doi.org/10.1007/s00222-009-0189-3
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Julius
Borcea and Petter
Brändén, The Lee-Yang and Pólya-Schur programs.
II. Theory of stable polynomials and applications, Comm. Pure Appl.
Math. 62 (2009), no. 12, 1595–1631. MR 2569072
(2011k:82026), http://dx.doi.org/10.1002/cpa.20295
- 7.
Julius
Borcea, Petter
Brändén, and Thomas
M. Liggett, Negative dependence and the geometry
of polynomials, J. Amer. Math. Soc.
22 (2009), no. 2,
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(2010b:62215), http://dx.doi.org/10.1090/S0894-0347-08-00618-8
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Brändén, Polynomials with the half-plane property and
matroid theory, Adv. Math. 216 (2007), no. 1,
302–320. MR 2353258
(2008h:05022), http://dx.doi.org/10.1016/j.aim.2007.05.011
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Petter
Brändén and David
G. Wagner, A converse to the Grace-Walsh-Szegő theorem,
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(13,727e)
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 lex/files/perma5.pdf
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Volker
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David
G. Wagner, Negatively correlated random variables and Mason’s
conjecture for independent sets in matroids, Ann. Comb.
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Wei, A criterion for the half-plane property, Discrete Math.
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(2010h:05076), http://dx.doi.org/10.1016/j.disc.2008.02.005
- 1.
- J. Borcea and P. Brändén, Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products, Duke Math. J. 143 (2008), 205-223. MR 2420507 (2009b:15015)
- 2.
- J. Borcea and P. Brändén, Lee-Yang problems and the geometry of multivariate polynomials, Lett. Math. Phys. 86 (2008), 53-61. MR 2460727 (2010e:82038)
- 3.
- J. Borcea and P. Brändén, Pólya-Schur master theorems for circular domains and their boundaries, Ann. of Math. 170 (2009), 465-492. MR 2521123 (2010g:30004)
- 4.
- J. Borcea and P. Brändén, Multivariate Pólya-Schur classification problems in the Weyl algebra, Proc. London Math. Soc. 101 (2010), 73-104.
- 5.
- J. Borcea and P. Brändén, The Lee-Yang and Pólya-Schur programs I: Linear operators preserving stability, Invent. Math. 177 (2009), 541-569. MR 2534100
- 6.
- J. Borcea and P. Brändén, The Lee-Yang and Pólya-Schur programs II: Theory of stable polynomials and applications Comm. Pure Appl. Math. 62 (2009), 1595-1631. MR 2569072
- 7.
- J. Borcea, P. Brändén, and T.M. Liggett, Negative dependence and the geometry of polynomials, J. Amer. Math. Soc. 22 (2009), 521-567. MR 2476782 (2010b:62215)
- 8.
- P. Brändén, Polynomials with the half-plane property and matroid theory, Adv. Math. 216 (2007), 302-320. MR 2353258 (2008h:05022)
- 9.
- P. Brändén and D.G. Wagner, A converse to the Grace-Walsh-Szegő theorem, Math. Proc. Camb. Phil. Soc. 147 (2009), 447-453. MR 2525937
- 10.
- Y.-B. Choe, J.G. Oxley, A.D. Sokal, and D.G. Wagner, Homogeneous polynomials with the half-plane property, Adv. in Appl. Math. 32 (2004), 88-187. MR 2037144 (2005d:05043)
- 11.
- L. Gårding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957-965. MR 0113978 (22:4809)
- 12.
- L. Gurvits, Van der Waerden/Schrijver-Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: one theorem for all. With a corrigendum, Electron. J. Combin. 15 (2008), R66 (26 pp). MR 2411443 (2009e:15015)
- 13.
- F.R. Harvey and H.B. Lawson Jr., Hyperbolic polynomials and the Dirichlet problem, http://arxiv.org/abs/0912.5220.
- 14.
- G. Hardy, J.E. Littlewood, and G. Pólya, ``Inequalities (Second Edition),'' Cambridge University Press, Cambridge, 1952. MR 0046395 (13:727e)
- 15.
- M. Laurent and A. Schrijver, On Leonid Gurvits' proof for permanents, http://homepages.cwi.nl/lex/files/perma5.pdf
- 16.
- Q.I. Rahman and G. Schmeisser, ``Analytic Theory of Polynomials,'' London Math. Soc. Monographs (N.S.) 26, Oxford University Press, New York, 2002. MR 1954841 (2004b:30015)
- 17.
- V. Scheidemann, ``Introduction to Complex Analysis in Several Variables,'' Birkhäuser, Basel, 2005. MR 2176976 (2006i:32001)
- 18.
- D.G. Wagner, Negatively correlated random variables and Mason's conjecture for independent sets in matroids, Ann. of Combin. 12 (2008), 211-239. MR 2428906 (2009f:05053)
- 19.
- D.G. Wagner and Y. Wei, A criterion for the half-plane property, Discrete Math. 309 (2009), 1385-1390. MR 2510546 (2010h:05076)
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Additional Information
David G. Wagner
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email:
dgwagner@math.uwaterloo.ca
DOI:
http://dx.doi.org/10.1090/S0273-0979-2010-01321-5
PII:
S 0273-0979(2010)01321-5
Received by editor(s):
May 17, 2010
Received by editor(s) in revised form:
June 12, 2010
Posted:
October 4, 2010
Additional Notes:
Research supported by NSERC Discovery Grant OGP0105392.
Dedicated:
In memoriam of Julius Borcea
Article copyright:
© Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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