Abstract
We introduce several notions of ‘random f-nomials’, i.e., random polynomials with a fixed number f of monomials of degree = ≤N The f exponents are chosen at random and then the coefficients are chosen to be Gaussian random, mainly from the SU(m + 1) ensemble. The results give limiting formulas as N→∞ for the (normalized) expected distribution of complex zeros of a system of k random f-nomials in m variables (k≤ m). When k = m, for SU(m+ 1) polynomials, the limit is the Monge-Ampère measure of a toric Kähler potential on ℂℙm obtained by averaging a ‘discrete Legendre transform’ of the Fubini-Study symplectic potential at f points of the unit simplex ∑ ⊂ ℝ
Mathematics Subject Classification (2000). Primary 32A60,60D05; Secondary 12D10, 14Q99.
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References
D.J. Bates, F. Bihan and F. Sottile, Bounds on the number of real solutions to polynomial equations. Int. Math. Res. Not. 2007, no. 23, Art. ID rnm114, 7 pp.
E. Bedford and B.A. Taylor, A new capacity for plurisubharmonic functions. Acta Math. 149 (1982), 1–40.
D.N. Bernstein, The number of roots of a system of equations. Functional Anal. Appl. 9 (1975), 183–185.
F. Bihan, J.M. Rojas and F. Sottile, On the sharpness of fewnomial bounds and the number of components of fewnomial hypersurfaces. Algorithms in algebraic geometry, 15–20, IMA Vol. Math. Appl., 146, Springer, New York, 2008.
P. Bleher and X. Di, Correlations between zeros of a random polynomial. J. Stat. Phys. 88 (1997), 269–305.
T. Bloom and B. Shiffman, Zeros of random polynomials on ℂm. Math. Res. Lett. 14 (2007), 469–479.
P. Bürgisser, Average Euler characteristic of random real algebraic varieties. C. R. Math. Acad. Sci. Paris 345 (2007), no. 9, 507–512.
D. Gayet and J.-Y.Welschinger, Exponential rarefaction of real curves with many components (arXiv:1005.3228v1). To appear in Pub. IHES.
J.M. Hammersley, The zeros of a random polynomial. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. II, 89–111, University of California Press, Berkeley and Los Angeles, 1956.
B. Ya. Kazarnovskii, On zeros of exponential sums. (Russian) Dokl. Akad. Nauk SSSR 257 (1981), no. 4, 804–808.
B. Ya. Kazarnovskii, Newton polyhedra and roots of systems of exponential sums. (Russian) Funktsional. Anal. i Prilozhen. 18 (1984), no. 4, 40–49, 96.
A.G. Khovanskii, Fewnomials, Trans. Math Monographs 88, AMS Publications, Providence, RI, 1991.
M. Klimek, Pluripotential Theory, London Math. Soc. Monographs, New Series 6, Oxford University Press, New York, 1991.
A.G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor. Invent. Math. 32 (1976), 1–31.
J.M. Rojas, On the average number of real roots of certain random sparse polynomial systems. The mathematics of numerical analysis (Park City, UT, 1995), 689–699, Lectures in Appl. Math. 32, Amer. Math. Soc., Providence, RI, 1996.
B. Shiffman, T. Tate and S. Zelditch, Distribution laws for integrable eigenfunctions. Ann. Inst. Fourier 54 (2004), no. 5, 1497–1546.
B. Shiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles. Comm. Math. Phys. 200 (1999), 661–683.
B. Shiffman and S. Zelditch, Random polynomials with prescribed Newton polytope. J. Amer. Math. Soc. 17 (2004), no. 1, 49–108.
B. Shiffman and S. Zelditch, Number variance of random zeros on complex manifolds. Geom. Funct. Anal. 18, No. 4 (2008), 1422–1475.
B. Shiffman, S. Zelditch and Q. Zhong, Random zeros on complex manifolds: conditional expectations. J. Inst. Math. Jussieu 10 (2011), 753–783.
M. Shub and S. Smale, Complexity of Bezout’s theorem. II. Volumes and probabilities. Computational algebraic geometry (Nice, 1992), 267–285, Progr. Math., 109, Birkhäuser Boston, Boston, MA, 1993.
J. Song and S. Zelditch, Convergence of Bergman geodesics on CP 1. Ann. Inst. Fourier 57, no. 6 (Festival Colin de Verdière) (2007), 2209–2237.
J. Song and S. Zelditch, Bergman metrics and geodesics in the space of Kähler metrics on toric varieties. Analysis & PDE 3, No. 2 (2010), 295–358.
J. Song and S. Zelditch, Test configurations, large deviations and geodesic rays on toric varieties (arXiv:0712.3599).
F. Sottile, Enumerative real algebraic geometry. Algorithmic and quantitative real algebraic geometry (Piscataway, NJ, 2001), 139–179, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 60, Amer. Math. Soc., Providence, RI, 2003.
B. Sturmfels, On the number of real roots of a sparse polynomial system. Hamiltonian and gradient flows, algorithms and control, Fields Inst. Commun. 3, Amer. Math. Soc., Providence, RI, 1994, pp. 137–143.
B. Sturmfels, Polynomial equations and convex polytopes. Amer. Math. Monthly 105 (1998), no. 10, 907–922.
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Shiffman, B., Zelditch, S. (2011). Random Complex Fewnomials, I. In: Brändén, P., Passare, M., Putinar, M. (eds) Notions of Positivity and the Geometry of Polynomials. Trends in Mathematics. Springer, Basel. https://doi.org/10.1007/978-3-0348-0142-3_20
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