Random Matrices and Random Algebraic Polynomials

Bharucha-Reid, A. T.; Sambandham, M.

doi:10.1016/b978-0-12-095710-1.50009-7

Cited by 21 publications

(25 citation statements)

References 9 publications

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“…Random binary and Gaussian coefficients are frequently studied for instance. As has been noted in [4], the statistical characteristics of the real roots of random polynomials do not seem to be heavily dependent on the fine details of the statistics of the coefficients.…”

Section: Definition 1 the Set Of Monic Uniform Random Polynomialsmentioning

confidence: 89%

Search decisions in a game of polynomial root counting

Raghunathan

Baillieul

2010

Proceedings of the 2010 American Control Conference

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There is increasing interest in having mixed teams of humans and smart machines carry out missions of search, surveillance, and reconnaissance. Understanding the way humans make decisions in such search problems is important to achieve effective cooperation. We report research on search styles we have observed in one dimensional search for real roots of randomly generated polynomials. The objective (solution) of the game is to determine the number of distinct real roots in [−1, 1], given an expression for the polynomial and a tool to sample the polynomial. Each player chooses a sequence of sample points in an attempt to collect sufficient information to estimate the number of roots in the interval. Data on eighteen subject players was collected and analyzed. The subjects were classified on the basis of three different types of player strategies that were identified in the games that the subjects played. A formal definition of indistinguishable players is given. Each of the subjects was modeled by a first-order Markov chain that is constructed using the recorded choices that were made by the subject. Using these Markov models, we have created computer programs that play the game so as to be indistinguishable (with respect to a predefined statistic) from the player that was modeled. Formal definitions of player confidence and boredom are proposed based on the analysis of the game play and resulting Markov model.

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Section: Definition 1 the Set Of Monic Uniform Random Polynomialsmentioning

confidence: 89%

Search decisions in a game of polynomial root counting

Raghunathan

Baillieul

2010

Proceedings of the 2010 American Control Conference

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“…, n , n ∈ N we obtain the random monic algebraic polynomials, (1.1). Trigonometric, hyperbolic and other types of random polynomials [5] can be obtained by suitably defining φ and f k . In the following we will often omit the qualifier generalized and call the objects defined by Eq.…”

Section: Introductionmentioning

confidence: 99%

“…and higher, m-point, correlation functions for the roots. Substituting (1.12) into (1.14) and introducing the joint distribution function of the values of the function and its derivative at the point z = x + iy 16) we see that the average density of roots at the point z = x + iy is given by the Kac-Rice [4,5,20] formula…”

Section: Introductionmentioning

confidence: 99%

Distribution of roots of random real generalized polynomials

et al. 1997

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The average density of zeros for monic generalized polynomials, $P_n(z)=\phi(z)+\sum_{k=1}^nc_kf_k(z)$, with real holomorphic $\phi ,f_k$ and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like $|\hbox{\rm Im}\,z|$. We present the low and high disorder asymptotic behaviors. Then we particularize to the large $n$ limit of the average density of complex roots of monic algebraic polynomials of the form $P_n(z) = z^n +\sum_{k=1}^{n}c_kz^{n-k}$ with real independent, identically distributed Gaussian coefficients having zero mean and dispersion $\delta = \frac 1{\sqrt{n\lambda}}$. The average density tends to a simple, {\em universal} function of $\xi={2n}{\log |z|}$ and $\lambda$ in the domain $\xi\coth \frac{\xi}{2}\ll n|\sin \arg (z)|$ where nearly all the roots are located for large $n$.Comment: 17 pages, Revtex. To appear in J. Stat. Phys. Uuencoded gz-compresed tarfile (.66MB) containing 8 Postscript figures is available by e-mail from mezin@spht.saclay.cea.f

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“…Our aim is to analyze, using the deterministic machinery of the previous sections, the probability that such a random power series is a Szegő power series. We point out to [3] and [10] as general references on random polynomials and on random power series.…”

Section: Random Power Seriesmentioning

confidence: 99%

Zeros of Sections of Power Series: Deterministic and Random

Fernández

2017

Comput. Methods Funct. Theory

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Abstract. We present a streamlined proof (and some refinements) of a characterization (due to F. Carlson and G. Bourion, and also to P. Erdős and H. Fried) of the so called Szegő power series. This characterization is then applied to readily obtain some (more) recent known results and some new results on the asymptotic distribution of zeros of sections of random power series, extricating quite naturally the deterministic ingredients. Finally, we study the possible limits of the zero counting probabilities of a power series.

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Random Matrices and Random Algebraic Polynomials

Cited by 21 publications

References 9 publications

Search decisions in a game of polynomial root counting

Search decisions in a game of polynomial root counting

Distribution of roots of random real generalized polynomials

Zeros of Sections of Power Series: Deterministic and Random

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