Random polynomials having few or no real zeros |
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Authors: | Amir Dembo Bjorn Poonen Qi-Man Shao Ofer Zeitouni |
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Affiliation: | Department of Mathematics & Statistics, Stanford University, Stanford, California 94305 ; Department of Mathematics, University of California, Berkeley, California 94720-3840 ; Department of Mathematics, University of Oregon, Eugene, Oregon 97403 ; Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel |
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Abstract: | Consider a polynomial of large degree whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly real zeros with probability as through integers of the same parity as the fixed integer . In particular, the probability that a random polynomial of large even degree has no real zeros is . The finite, positive constant is characterized via the centered, stationary Gaussian process of correlation function . The value of depends neither on nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability one may specify also the approximate locations of the zeros on the real line. The constant is replaced by in case the i.i.d. coefficients have a nonzero mean. |
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Keywords: | Random polynomials Gaussian processes |
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