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Polynomial Roots and Calabi-Yau Geometries

DOI: 10.1155/2011/719672

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The examination of roots of constrained polynomials dates back at least to Waring and to Littlewood. However, such delicate structures as fractals and holes have only recently been found. We study the space of roots to certain integer polynomials arising naturally in the context of Calabi-Yau spaces, notably Poincaré and Newton polynomials, and observe various salient features and geometrical patterns. 1. Introduction and Summary The subject of roots of monovariate polynomials is, without doubt, an antiquate one and has germinated an abundance of fruitful research over the ages. It is, therefore, perhaps surprising that any new statements could at all be made regarding such roots. The advent of computer algebra, chaotic phenomena, and random ensembles has, however, indeed shed new light upon so ancient a metier. Polynomials with constrained coefficients and form, though permitted to vary randomly, have constituted a vast field itself. As far back as 1782, Edward Waring, in relation to his famous problem on power summands, had shown that for cubic polynomials with random real coefficients, the ratio of the probability of finding nonreal zeros versus that of not finding non-real zeros is less than or equal to 2. Constraining the coefficients to be integers within a fixed range has, too, its own history. It was realised in [1] that a degree random polynomial with distributed evenly, the expected number of real roots is of order asymptotically in . This was furthered by [2] to be essentially independent of the statistics, in that has the same asymptotics (cf. also [3, 4]), as much for being evenly distributed real numbers, in , or as Gaussian distributed in . Continual development ensued (q.v. also [5]), notably by Littlewood [6], Erd?s and Turán [7], Hammersley [8], and Kac [9]. Indeed, a polynomial with coefficients only taking values as has come to be known as a Littlewood polynomial, and the Littlewood Problem asks for the the precise asymptotics, in the degree, of such polynomials taking values, with complex arguments, on the unit circle. The classic work of Montgomery [10] and Odlyzko [11], constituting one of the most famous computer experiments in mathematics (q. v. Section?? 3.1 of [12] for some recent remarks on the distributions), empirically showed that the distribution of the (normalized) spacings between successive critical zeros of the Riemann zeta function is the same as that of a Gaussian unitary ensemble of random matrices, whereby infusing our subject with issues of uttermost importance. Subsequently, combining the investigation of zeros


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