Multivariate Pólya-Schur classification problems in the Weyl algebra

Proceedings of the London Mathematical Society

Volumn 101, Issue 1, 2010, Pages 73-104

  Borcea, J ^a   Branden P ^a,b  

^a STOCKHOLM UNIVERSITY   (Sweden)

^b ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77956021403     PISSN: 00246115     EISSN: 1460244X     Source Type: Journal    
DOI: 10.1112/plms/pdp049     Document Type: Article

References (56)

1. A. Aleman, D. Beliaev and H. Hedenmalm, 'Real zero polynomials and Ṕolya-Schur type theorems', J. Anal. Math. 94 (2004) 49-60.
2. M. F. Atiyah, R. Bott and L. G? arding, 'Lacunas for hyperbolic differential operators with constant coefficients I', Acta Math. 124 (1970) 109-189.
3. V. Bargmann, 'On a Hilbert space of analytic functions and an associated integral transform, Part I', Comm. Pure Appl. Math. 14 (1961) 187-214.
4. V. Bargmann, 'Remarks on a Hilbert space of analytic functions', Proc. Natl. Acad. Sci. USA 48 (1962) 199-204.
5. B. Beauzamy, E. Bombieri, P. Enflo and H. L. Montgomer, 'Products of polynomials in many variables', J. Number Theory 36 (1990) 219-245.
6. J.-E. Björk, Rings of differential operators, North-Holland Mathematical Library 21 (North-Holland, Amsterdam, 1979).
7. J. Borcea, 'Spectral order and isotonic differential operators of Laguerre-Ṕolya type', Ark. Mat. 44 (2006) 211-240.
8. J. Borcea and P. Branden, 'Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products', Duke Math. J. 143 (2008) 205-223.
9. J. Borcea and P. Branden, 'Ṕolya-Schur master theorems for circular domains and their boundaries', Ann. of Math. (2) 170 (2009) 465-492.
10. J. Borcea, P. Brand́en, G. Csordas and V. Vinnikov, 'Ṕolya-Schur-Lax problems: hyperbolicity and stability preservers', http://www.aimath.org/pastworkshops/polyaschurlax.html.
11. J. Borcea, P. Brand́en and T. M. Liggett, 'Negative dependence and the geometry of polynomials', J. Amer. Math. Soc. 22 (2009) 521-567.
12. N. G. de Bruijn, 'Some theorems on the roots of polynomials', Nieuw Arch. Wiskd. 23 (1949) 66-68.
13. N. G. de Bruijn, 'An analogue of Grace's apolarity theorem', Nieuw Arch. Wiskd 23 (1949) 69-76.
14. P. Brand́en, 'On linear transformations preserving the Ṕolya frequency property', Trans. Amer. Math. Soc. 358 (2006) 3697-3716.
15. P. Brand́en, 'Polynomials with the half-plane property and matroid theory', Adv. Math. 216 (2007) 302-320.
16. Y. Choe, J. Oxley, A. Sokal and D. Wagner, 'Homogeneous multivariate polynomials with the halfplane property', Adv. Appl. Math. 32 (2004) 88-187.
17. T. Craven and G. Csordas, 'On the number of real roots of polynomials', Pacific J. Math. 102 (1982) 15-28.
18. T. Craven and G. Csordas, 'Composition theorems, multiplier sequences and complex zero decreasing sequences', Value distribution theory and its related topics, Advanced Complex Analysis of Application 3 (Reviewer: Arturas Dubickas; Kluwer Press, Boston, 2004) 131-166.
19. G. Csordas, 'Linear operators and the distribution of zeros of entire functions', Complex Var. Elliptic Equ. 51 (2006) 625-632.
20. P. A. M. Dirac, 'La seconde quantification', Ann. Inst. H. Poincaŕe 11 (1949) 15-47.
21. E. Fischer, 'Uber algebraische Modulsysteme und lineare homogene partielle Differentialgleichungen mit konstanten Koeffizienten', J. reine angew. Math. 140 (1911) 48-81.
22. E. Fischer, 'Uber die Differentiationsprozesse der Algebra', J. reine angew. Math. 148 (1917) 1-78.
23. V. A. Fock, 'Verallgemeinerung und Lösnung der Diracschen statistichen Gleichung', Z. Phys. 49 (1928) 339-357.
24. V. A. Fock, 'Zur Quantenelektrodynamik', Phys. Zeit. Sowjetunion 6 (1934) 425-469.
25. F. R. Gantmacher, The theory of matrices, vol.1. (Chelsea Publishing, Providence, RI, 1998).
26. L. Garding, 'An inequality for hyperbolic polynomials', J. Math. Mech. 8 (1959) 957-965.
27. C. D. Godsil, Algebraic combinatorics, (Chapman & Hall, New York, 1993).
28. L. Gurvits, 'Hyperbolic polynomials approach to van der Waerden and Schrijver-Valiant like conjectures: sharper bounds, simpler proofs and algorithmic applications', Proceeding of the STOC'06, (ACM, New York, 2006) 417-426.
29. O. J. Heilmann and E. H. Lieb, 'Theory of monomer-dimer systems', Comm. Math. Phys. 25 (1972) 190-232.
30. J. Helton and V. Vinnikov, Linear matrix inequality representation of sets, Comm. Pure Appl. Math. 60 (2007) 654-674.
31. A. Hinkkanen, 'Schur products of certain polynomials', Lipa's Legacy: Proceedings of the Bers Colloquium (eds. J. Dodziuk and L. Keenin) Contemporous Mathematics 211 (American Mathematical Society, Providence, RI, 1997) 285-295.
32. L. Hörmander, Notions of convexity, Progress in Mathematics 127 (Birkhauser, Boston, MA, 1994).
33. V. Kharitonov and J. Torres Muñoz, 'Robust stability of multivariate polynomials. Part I: small coefficient perturbations', Multidimens. Syst. Signal Process. 10 (1999) 7-20.
34. E. Laguerre, 'Fonctions du genre źero et du genre un', C. R. Acad. Sci. Paris 95 (1882) 828-831.
35. P. D. Lax, 'Differential equations, difference equations and matrix theory', Comm. Pure Appl. Math. 6 (1958) 175-194.
36. T. Lee and C. Yang, 'Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising models', Phys. Rev. 87 (1952) 410-419.
37. B. Ja. Levin, Distribution of zeros of entire functions'. Translations of Mathematical Monographs 5 (American Mathematical Society, Providence, RI, 1980).
38. A. Lewis, P. Parillo and M. Ramana, 'The Lax conjecture is true', Proc. Amer. Math. Soc. 133 (2005) 2495-2499.
39. E. H. Lieb and A. D. Sokal, 'A General Lee-Yang theorem for one-component and multicomponent ferromagnets', Comm. Math. Phys. 80 (1981) 153-179.
40. B. Malgrange, Equations diff́erentielles ̀a coefficients polynomiaux, Progress in Mathematics 96 (Birkhauser, Boston, 1991) (French).
41. M. Marden, The geometry of the zeros of a polynomial in a complex variable, Mathematical Surveys of Monograrphs 3 (American Mathematical Society, New York, 1949).
42. D. J. Newman and H. S. Shapiro, A Hilbert space of entire functions related to the operational calculus, Mimeographed Notes (University of Michigan, Ann Arbor, 1964) 92pp.
43. D. J. Newman and H. S. Shapiro, 'Certain Hilbert spaces of entire functions', Bull. Amer. Math. Soc. 72 (1966) 971-977.
44. D. J. Newman and H. S. Shapiro, 'Fischer spaces of entire functions', 1968 Entire functions and related parts of analysis, Proceedings of Symposia in Pure Mathematics 11 (American Mathematical Society, Providence, RI, 1968) 360-369.
45. N. Obreschkoff, Verteilung und Berechnung der Nullstellen Reeller Polynome (VEB Deutscher Verlag der Wissenschaften, Berlin, 1963).
46. V. Olshevsky and L. Sakhnovich, 'A generalized Kharitonov theorem for quasi-polynomials and entire functions occurring in systems with multiple and distributed delays', Advanced signal processing algorithms, architectures, and implementations XV, (ed. F. T. Luk; SPIE Publications, 2005).
47. J. Peetre, 'Une caract́erisation abstraite des oṕerateurs diff́erentiels', Math. Scand. 7 (1959) 211-218; Erratum, Math. Scand. 8 (1960) 116-120.
48. G. Ṕolya, 'Uber algebraische Gleichungen mit nur reellen Wurzeln', Vierteljschr. Naturforsch. Ges. ZUrich 61 (1916) 546-548.
49. G. Ṕolya and J. Schur, 'Uber zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen', J. reine angew. Math. 144 (1914) 89-113.
50. G. Ṕolya and G. Szegö, Problems and theorems in analysis, vol.II (Springer, Berlin, 1976).
51. Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, London Mathematical Society Monographs (New Series) 26 (Oxford University Press, New York, 2002).
52. J. Renegar, 'Hyperbolic programs, and their derivative relaxations', Found. Comput. Math. 6 (2006) 59-79.
53. B. Reznick, 'An inequality for products of polynomials', Proc. Amer. Math.Soc. 117 (1993) 1063-1073.
54. I. Schur, 'Zwei Satze Uber algebraische Gleichungen mit lauter reellen Wurzeln', J. reine angew. Math. 144 (1923) 75-88.
55. I. E. Segal, 'Mathematical characterization of the physical vacuum for a linear Bose-Einstein field', Illinois J. Math. 6 (1962) 500-523.
56. H. S. Shapiro, 'Spectral aspects of a class of differential operators', Operator methods in ordinary and partial differential equations, Stockholm, 2000, Operator Theory: Advances and Application 132, (Birkhauser, Basel, 2002) 361-385.