Homogeneous multivariate polynomials with the half-plane property

Advances in Applied Mathematics

Volumn 32, Issue 1-2, 2004, Pages 88-187

  Choe, Young Bin ^a   Oxley, James G ^b   Sokal, Alan D ^c   Wagner, David G ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

^b LOUISIANA STATE UNIVERSITY   (United States)

^c POLYTECHNIC UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0346401580     PISSN: 01968858     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0196-8858(03)00078-2     Document Type: Article

References (80)

1. L.V. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1966.
2. N. Anderson, E.B. Saff, R.S. Varga, On the Eneström-Kakeya theorem and its sharpness, Linear Algebra Appl. 28 (1979) 5-16.
3. T. Asano, Theorems on the partition functions of the Heisenberg ferromagnets, J. Phys. Soc. Japan 29 (1970) 350-359.
4. R.B. Bapat, T.E.S. Raghavan, Nonnegative Matrices and Applications, Cambridge Univ. Press, Cambridge, 1997.
5. S. Basu, R. Pollack, M.-F. Roy, On the combinatorial and algebraic complexity of quantifier elimination, J. ACM 43 (1996) 1002-1045.
6. J. Bokowski, B. Sturmfels, Computational Synthetic Geometry, in: Lecture Notes in Math., Vol. 1355, Springer-Verlag, Berlin, 1989.
7. N.K. Bose, A note on multivariable positive real functions, IEEE Trans. Circuit Theory CT-19 (1972) 194-195.
8. A. Bouchet, W.H. Cunningham, Delta-matroids, jump systems, and bisubmodular polyhedra, SIAM J. Discrete Math. 8 (1995) 17-32.
9. R.L. Brooks, C.A.B. Smith, A.H. Stone, W.T. Tutte, The dissection of rectangles into squares, Duke Math. J. 7 (1940) 312-340.
10. J.I. Brown, C.J. Colbourn, Roots of the reliability polynomial, SIAM J. Discrete Math. 5 (1992) 571-585.
11. R.A. Brualdi, Transversal matroids, Chapter 5 in: N. White (Ed.), Combinatorial Geometries, in: Encyclopedia Math. Appl., Vol. 29, Cambridge Univ. Press, Cambridge, 1987, pp. 72-97.
12. O. Brune, Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency, J. Math. Phys. 10 (1931) 191-236.
13. T.H. Brylawski, An affine representation for transversal geometries, Stud. Appl. Math. 54 (1975) 143-160.
14. T. Brylawski, Constructions, Chapter 7 in: N. White (Ed.), Theory of Matroids, in: Encyclopedia Math. Appl., Vol. 26, Cambridge Univ. Press, Cambridge, 1986, pp. 127-223.
15. B.F. Caviness, J.R. Johnson (Eds.), Quantifier Elimination and Cylindrical Algebraic Decomposition, Springer-Verlag, Wien, 1998.
16. S. Chaiken, A combinatorial proof of the all minors matrix-tree theorem, SIAM J. Algebra Discrete Meth. 3 (1982) 319-329.
17. S. Chaiken, D.J. Kleitman, Matrix tree theorems, J. Combin. Theory A 24 (1978) 377-381.
18. W.-K. Chen, Applied Graph Theory, North-Holland, Amsterdam, 1971.
19. C.J. Colbourn, The Combinatorics of Network Reliability, Oxford Univ. Press, New York, 1987.
20. G.E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, in: H. Brakhage (Ed.), Automata Theory and Formal Languages, in: Lecture Notes in Comput. Sci., Vol. 33, Springer-Verlag, Berlin, 1975, pp. 134-183; reprinted in: B.F. Caviness, J.R. Johnson (Eds.), Quantifier Elimination and Cylindrical Algebraic Decomposition, Springer-Verlag, Wien, 1998, pp. 85-121.
21. W.H. Cunningham, A combinatorial decomposition theoxy, PhD thesis, University of Waterloo, 1973.
22. D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs: Theory and Applications, Academic Press, New York, 1980.
23. C.A. Desoer, E.S. Kuh, Basic Circuit Theory, McGraw-Hill, New York, 1969.
24. R.J. Duffin, T.D. Morley, Wang algebra and matroids, IEEE Trans. Circuits Systems 25 (1978) 755-762.
25. A. Fettweis, Elementary symmetric functions and positive functions in several variables, in: M.A. Kaashoek, J.H. van Schuppen, A.C.M. Ran (Eds.), Signal Processing, Scattering and Operator Theory, and Numerical Methods, Birkhäuser, Boston, 1990, pp. 91-98.
26. A. Fettweis, S. Basu, New results on stable multidimensional polynomials - Part I: Continuous case, IEEE Trans. Circuits Systems 34 (1987) 1221-1232.
27. F.R. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New York, 1959.
28. J.F. Geelen, Lectures on jump systems, Center of Parallel Computing, University of Cologne, n.d.
29. J.F. Geelen, A.M.H. Gerards, A. Kapoor, The excluded minors for GF(4)-representable matroids, J. Combin. Theory B 79 (2000) 247-299.
30. J.F. Geelen, J.G. Oxley, D.L. Vertigan, G.P. Whittle, On the excluded minors for quaternary matroids, J. Combin. Theory B 80 (2000) 57-68.
31. J.H. Grace, The zeros of a polynomial, Proc. Cambridge Philos. Soc. 11 (1902) 352-357.
32. D.H. Greene, D.E. Knuth, Mathematics for the Analysis of Algorithms, 3rd edition, Birkhäuser, Boston, 1990.
33. J. Gregor, On quadratic Hurwitz forms I, Aplikace Matematiky 26 (1981) 142-153.
34. D.Yu. Grigor'ev, Complexity of deciding Tarski algebra, J. Symbolic Comput. 5 (1988) 65-108.
35. O.J. Heilmann, E.H. Lieb, Theory of monomer-dimer systems, Commun. Math. Phys. 25 (1972) 190-232.
36. J. Heintz, M.F. Roy, P. Solernó, Sur la complexité du principe de Tarski-Seidenberg, Bull. Soc. Math. France 118 (1990) 101-126.
37. A. Hinkkanen, Schur products of certain polynomials, in: J. Dodziuk, L. Keen (Eds.), Lipa's Legacy: Proceedings of the Bers Colloquium, in: Contemp. Math., Vol. 211, Amer. Math. Society, Providence, RI, 1997, pp. 285-295.
38. A.J. Hoffman, Eigenvalues and partitionings of the edges of a graph, Linear Algebra Appl. 5 (1972) 137-146.
39. T.W. Hungerford, Algebra, Springer-Verlag, New York, 1974, fifth printing 1989.
40. A.W. Ingleton, Transversal matroids and related structures, in: M. Aigner (Ed.), Higher Combinatorics, Reidel, Dordrecht, 1977, pp. 117-131.
41. T. Koga, Synthesis of finite passive n-ports with prescribed positive real matrices of several variables, IEEE Trans. Circuit Theory CT-15 (1968) 2-23.
42. S.G. Krantz, Function Theory of Several Complex Variables, 2nd edition, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1992.
43. S. Krogdahl, The dependence graph for bases in matroids, Discrete Math. 19 (1977) 47-59.
44. J.P.S. Kung, Basis-exchange properties, Chapter 4 in: N. White (Ed.), Theory of Matroids, in: Encyclopedia Math. Appl., Vol. 26, Cambridge Univ. Press, Cambridge, 1986, pp. 62-75.
45. J.P.S. Kung, Twelve views of matroid theory, in: S. Hong, J.H. Kwah, K.H. Kim, F.W. Roush (Eds.), Combinatorial and Computational Mathematics, Pohang, Korea, 15-17 February 2000, World Scientific, Singapore, 2001, pp. 56-96.
46. T.D. Lee, C.N. Yang, Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Phys. Rev. 87 (1952) 410-419.
47. E.H. Lieb, A.D. Sokal, A general Lee-Yang theorem for one-component and multicomponent ferromagnets, Commun. Math. Phys. 80 (1981) 153-179.
48. L. Lovász, M.D. Plummer, Matching Theory, in: North-Holland Math. Stud., Vol. 121, in: Ann. Discrete Math., Vol. 29, North-Holland, Amsterdam, 1986.
49. L. Loväsz, The membership problem in jump systems, J. Combin. Theory B 70 (1997) 45-66.
50. M. Marcus, H. Minc, Introduction to Linear Algebra, Macmillan, New York, 1965.
51. M. Marden, Geometry of Polynomials, 2nd edition, Amer. Math. Society, Providence, RI, 1966.
52. J.H. Mason, A characterization of transversal independence spaces, in: C.P. Bruter (Ed.), Théorie des Matroïdes, in: Lecture Notes in Math., Vol. 211, Springer-Verlag, Berlin, 1971, pp. 86-94.
53. H. Minc, Permanents, Addison-Wesley, Reading, MA, 1978.
54. C.M. Newman, Zeros of the partition function for generalized Ising systems, Comm. Pure Appl. Math. 27 (1974) 143-159.
55. J.G. Oxley, Matroid Theory, Oxford Univ. Press, New York, 1992.
56. J. Oxley, C. Semple, D. Vertigan, Generalized Δ-Y exchange and k-regular matroids, J. Combin. Theory B 79 (2000) 1-65.
57. H. Ozaki, T. Kasami, Positive real functions of several variables and their applications to variable networks, IEEE Trans. Circuit Theory CT-7 (1960) 251-260.
58. P. Penfield, R. Spence, S. Duinker, Tellegen's Theorem and Electrical Networks, MIT Press, Cambridge, MA, 1970.
59. M.J. Piff, D.J.A. Welsh, On the vector representation of matroids, J. London Math. Soc. (2) 2 (1970) 284-288.
60. J. Renegar, On the computational complexity and geometry of the first-order theory of the reals. I-III, J. Symbolic Comput. 13 (1992) 255-352.
61. G. Royle, A.D. Sokal, The Brown-Colbourn conjecture on zeros of reliability polynomials is false, in preparation.
62. W. Rudin, Function Theory in Polydiscs, Benjamin, New York, 1969.
63. D. Ruelle, Extension of the Lee-Yang circle theorem, Phys. Rev. Lett. 26 (1971) 303-304, 870, Erratum.
64. D. Ruelle, Some remarks on the location of zeroes of the partition function for lattice systems, Commun. Math. Phys. 31 (1973) 265-277.
65. C. Semple, k-regular matroids, PhD thesis, Victoria University, Wellington, 1998.
66. C. Semple, On maximum-sized k-regtdar matroids, Graphs Combin. 15 (1999) 441-462.
67. C. Semple, G. Whittle, Partial fields and matroid representation, Adv. Appl. Math. 17 (1996) 184-208.
68. B. Simon, The P( )2 Euclidean (Quantum) Field Theory, Section IX. 3, Princeton Univ. Press, Princeton, 1974.
69. J.H. Smith, Some properties of the spectrum of a graph, in: Combinatorial Structures and their Applications, Proceedings of the Calgary International Conference, Calgary, Alberta, 1969, Gordon and Breach, New York, 1970, pp. 403-406.
70. A.D. Sokal, Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions, Combin. Probab. Comput. 10 (2001) 41-77.
71. G. Szegö, Bemerkungen zu einem Satz von J.H. Grace über die Wurzeln algebraischer Gleichungen, Math. Z. 13 (1922) 28-55; reprinted in: R. Askey (Ed.), G. Szegö, Collected Papers, Vol. 1, Birkhäuser, Boston, 1982, pp. 506-534.
72. A. Tarski, A Decision Method for Elementary Algebra and Geometry, University of California Press, Berkeley, 1951; reprinted in: B.F. Caviness, J.R. Johnson (Eds.), Quantifier Elimination and Cylindrical Algebraic Decomposition, Springer-Verlag, Wien, 1998, pp. 24-84.
73. D.G. Wagner, Zeros of reliability polynomials and f-vectors of matroids, Combin. Probab. Comput. 9 (2000) 167-190.
74. J.L. Walsh, On the location of the roots of certain types of polynomials, Trans. Amer. Math. Soc. 24 (1922) 163-180.
75. N. White, Coordinatizations, Chapter 1 in: N. White (Ed.), Combinatorial Geometries, in: Encyclopedia Math. Appl., Vol. 29, Cambridge Univ. Press, Cambridge, 1987, pp. 1-27.
76. N. White, Unimodular matroids, Chapter 3 in: N. White (Ed.), Combinatorial Geometries, in: Encyclopedia Math. Appl., Vol. 29, Cambridge Univ. Press, Cambridge, 1987, pp. 40-52.
77. G. Whittle, A characterisation of the matroids representable over GF(3) and the rationals, J. Combin. Theory B 65 (1995) 222-261.
78. G. Whittle, On matroids representable over GF(3) and other fields, Trans. Amer. Math. Soc. 349 (1997) 579-603.
79. H.S. Wilf, Generatingfunctionology, 2nd edition, Academic Press, Boston, 1994.
80. D. Zeilberger, A combinatorial approach to matrix algebra, Discrete Math. 56 (1985) 61-72.