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  • Currently known as: Journal of the Australian Mathematical Society Title history
    Journal of the Australian Mathematical Society, Volume 85, Issue 1
  • August 2008, pp. 81-86

REAL ZEROS OF ALGEBRAIC POLYNOMIALS WITH STABLE RANDOM COEFFICIENTS

  • K. FARAHMAND (a1)
  • DOI: http://dx.doi.org/10.1017/S1446788708000682
  • Published online: 01 August 2008
Abstract
Abstract

We consider a random algebraic polynomial of the form Pn,θ,α(t)=θ0ξ0+θ1ξ1t+⋯+θnξntn, where ξk, k=0,1,2,…,n have identical symmetric stable distribution with index α, 0<α≤2. First, for a general form of θk,αθk we derive the expected number of real zeros of Pn,θ,α(t). We then show that our results can be used for special choices of θk. In particular, we obtain the above expected number of zeros when . The latter generate a polynomial with binomial elements which has recently been of significant interest and has previously been studied only for Gaussian distributed coefficients. We see the effect of α on increasing the expected number of zeros compared with the special case of Gaussian coefficients.

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[2]A. Edelman and E. Kostlan , ‘How many zeros of a random polynomial are real?’, Bull. Amer. Math. Soc. 32 (1995), 137.

[4]K. Farahmand and A. Nezakati , ‘Algebraic polynomials with non-identical random coefficients’, Proc. Amer. Math. Soc. 133 (2005), 275283.

[5]M. Kac , ‘On the average number of real roots of a random algebraic equation’, Bull. Amer. Math. Soc. 49 (1943), 314320.

[8]A. Ramponi , ‘A note on the complex roots of complex random polynomials’, Stat. Probab. Lett. 44 (1999), 181187.

[9]J. E. Wilkins , ‘An asymptotic expansion for the expected number of real zeros of a random polynomial’, Proc. Amer. Math. Soc. 103 (1988), 12491258.

[10]J. E. Wilkins , ‘Mean number of real zeros of a random trigonometric polynomial’, Proc. Amer. Math. Soc. 111 (1991), 851863.

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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