Equilibrium distribution of zeros of random polynomials

International Mathematics Research Notices

Volumn , Issue 1, 2003, Pages 25-49

  Shiffman, Bernard ^a   Zelditch, Steve ^a  

^a JOHNS HOPKINS UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038578410     PISSN: 10737928     EISSN: None     Source Type: Journal    
DOI: 10.1155/s1073792803206073     Document Type: Article

References (28)

1. S. R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping, Studies in Advanced Mathematics, CRC Press, Florida, 1992.
2. S. Bergman, The Kernel Function and Conformal Mapping, Mathematical Surveys, no. 5, American Mathematical Society, Rhode Island, 1970.
3. P. Bleher and X. Di, Correlations between zeros of a random polynomial, J. Statist. Phys. 88 (1997), no. 1-2, 269-305.
4. P. Bleher and D. Ridzal, SU(1, 1) random polynomials, J. Statist. Phys. 106 (2002), no. 1-2, 147-171.
5. P. Bleher, B. Shiffman, and S. Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142 (2000), no. 2, 351-395.
6. _, Correlations between zeros and supersymmetry, Comm. Math. Phys. 224 (2001), no. 1, 255-269.
7. T. Carleman, Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen, Ark. Mat. Astr. Fys. 17 (1923), no. 9, 1-29 (German).
8. P. A. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes in Mathematics, vol. 3, Courant Institute of Mathematical Sciences, New York University, New York, 1999.
9. A. Dembo, B. Poonen, Q.-M. Shao, and O. Zeitouni, Random polynomials having few or no real zeros, preprint, 2000, http://arxiv.org/abs/math.PR/0006113.
10. A. Edelman and E. Kostlan, How many zeros of a random polynomial are real? Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 1, 1-37.
11. P. Erdös and A. C. Offord, On the number of real roots of a random algebraic equation, Proc. London Math. Soc. (3) 6 (1956), 139-160.
12. P. Erdös and P. Turán, On the distribution of roots of polynomials, Ann. of Math. (2) 51 (1950), 105-119.
13. J. M. Hammersley, The zeros of a random polynomial, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. II, University of California Press, California, 1956, pp. 89-111.
14. J. H. Hannay, Chaotic analytic zero points: exact statistics for those of a random spin state, J. Phys. A 29 (1996), no. 5, L101-L105.
15. I. A. Ibragimov and N. B. Maslova, The mean number of real zeros of random polynomials. I. Coefficients with zero mean, Teor. Verojatnost. i Primenen. 16 (1971), 229-248 (Russian, English summary), English translation, Theor. Probability Appl. 16 (1971), 228-248.
16. M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49 (1943), 314-320.
17. _, On the average number of real roots of a random algebraic equation. II, Proc. London Math. Soc. (2) 50 (1949), 390-408.
18. J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation, J. London Math. Soc. 13 (1938), 288-295.
19. _, On the number of real roots of a random algebraic equation, II, Proc. Cambridge Philos. Soc. 35 (1939), 133-148.
20. _, On the number of real roots of a random algebraic equation. III, Rec. Math. [Mat. Sbornik] N.S. 12(54) (1943), 277-286.
21. L. A. Shepp and R. J. Vanderbei, The complex zeros of random polynomials, Trans. Amer. Math. Soc. 347 (1995), no. 11, 4365-4384.
22. B. Shiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys. 200 (1999), no. 3, 661-683.
23. _, Random polynomials with prescribed Newton polytope. I, preprint, 2002, http://arxiv.org/abs/math.AG/0203074.
24. V. I. Smirnov and N. A. Lebedev, Functions of a Complex Variable: Constructive Theory, The MIT Press, Massachusetts, 1968.
25. M. Sodin, Zeros of Gaussian analytic functions, Math. Res. Lett. 7 (2000), no. 4, 371-381.
26. P. K. Suetin, Polynomials orthogonal over a region and Bieberbach polynomials, Proceedings of the Steklov Institute of Mathematics, no. 100 (1971), American Mathematical Society, Rhode Island, 1974.
27. G. Szegö, Über orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene gehoren, Math. Zeit. 9 (1921), 218-270 (German).
28. G. Szegö, Orthogonal Polynomials, 4th ed., American Mathematical Society, Colloquium Publications, vol. 23, American Mathematical Society, Rhode Island, 1975.