On the Average Number of Real Roots of a Random Algebraic Equation

Farahmand, Kambiz

doi:10.1214/aop/1176992539

Cited by 35 publications

(37 citation statements)

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“…Their method allowed them [5] to show that if the means of all the coefficients are equal but nonzero, then the expected number of real roots asymptotically reduces by half. This remarkable reduction of the real roots by half persists in the work of Farahmand [2,3] when he studied the real roots of the equation T(x) = K for any nonzero K .…”

Section: Introductionmentioning

confidence: 90%

Real zeros of random algebraic polynomials

Farahmand¹

1991

Proc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 90%

Real zeros of random algebraic polynomials

Farahmand¹

1991

Proc. Amer. Math. Soc.

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“…The second result in Theorem 1.2 is not surprising when we consider the results, outlined earlier, from Farahmand [4]. From Kac [8] and Das [3] we know that for the intervals (−∞, −1) and (1, ∞) the expected number of both zero crossings and turning points are asymptotically equal.…”

Section: Theorem 12 If the Coefficients Of T (X) In (13) Are Indepmentioning

confidence: 55%

“…Kac [8] found the leading behaviour and an error term for the expected number of zero crossings of the random polynomial (1.3). Farahmand [4] found the expected number of K level crossings in the interval (−1, 1) is asymptotic to (1/π ) log(n/K 2 ), and in the intervals (−∞, −1)…”

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confidence: 99%

Local maxima of a random algebraic polynomial

Farahmand¹,

Hannigan²

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. We present a useful formula for the expected number of maxima of a normal process ξ(t) that occur below a level u. In the derivation we assume chiefly that ξ(t), ξ (t), and ξ (t) have, with probability one, continuous 1 dimensional distributions and expected values of zero. The formula referred to above is then used to find the expected number of maxima below the level u for the random algebraic polynomial. This result highlights the very pronounced difference in the behaviour of the random algebraic polynomial on the interval (−1, 1) compared with the intervals (−∞, −1) and (1, ∞). It is also shown that the number of maxima below the zero level is no longer O(log n) on the intervals (−∞, −1) and (1, ∞).2000 Mathematics Subject Classification. Primary 60H99, 26C99. Introduction.A significant amount has been written concerning the mean number of crossings of a fixed level, by both stationary and nonstationary normal processes. Farahmand [5] has given a formula for the expected number of maxima, below a level u, for a normal process under certain assumptions. The initial interest in the stationary case can be traced back to Rice [13], and was subsequently investigated by Ito [7] and Ylvisarer [15]. These works concentrated on level crossings and have been reviewed in the comprehensive book by Cramér and Leadbetter [2]. In this book, Cramér and Leadbetter have also given a formula for the expected number of local maxima. In addition, their method enabled them to find the distribution function for the height of a local maximum. Leadbetter [9], in his treatment of the nonstationary case, gives a result for the mean number of level crossings by a normal process. In his work Leadbetter assumes that the random process has continuous sample functions, with probability one.In what follows we consider ξ(t) to be a real valued normal process. We assume ξ(t), and its first and second derivatives ξ (t) and ξ (t), posses, with probability one, continuous one dimensional distributions, such that the mean number of crossings of any level by ξ(t), and the zero level by ξ (t) are finite. In addition, we assume that the mean of ξ(t), ξ (t), and ξ (t) are all zero. ξ(t) has a local maximum at t = t i , if ξ (t) has a down crossing of the level zero at t i . The local maxima which are of interest here, are those that occur when ξ(t) is also below the level u. The total number of down crossings of the level zero by ξ (t) in (α, β) is defined as M(α, β), and these occur at the points α < t 1 < t 2 < ··· < t M(α,β) < β. We define M u (α, β) as the number of zero down crossings by ξ (t), where 0 ≤ i ≤ M(α, β) and ξ(t i ) ≤ u. The corollary presented below is a corollary to Theorem 1.1 in [5]. The corollary is proved in Section 2.

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“…Denote by N K (α, β) the number of real roots of the equation P (x) = K in the interval (α, β) and by EN K (α, β) its expected value. In particular it is shown (for example see Kac [8] or Wilkins [12]) that if the coefficients are assumed to have a standard normal distribution and n is sufficiently large, EN 0 (−∞, ∞) ∼ (2/π) log n. Recently (see Farahmand [4]), it was shown that this asymptotic value remains valid for EN K (−∞, ∞) as long as K is bounded.…”

Section: Introductionmentioning

confidence: 99%

“…For K large such that K 2 /n −→ 0 as n −→ ∞, EN K (−∞, ∞) is asymptotically reduced to (1/π) log(n/K 2 ) in (−1, 1) while it remains the same as for K = 0 in (−∞, −1) ∪ (1, ∞) ( [4]). In contrast, a random trigonometric polynomial [5] and [6].…”

Section: Introductionmentioning

confidence: 99%