Hypergeometric polynomials are optimal
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Mathematische Zeitschrift
Hypergeometric polynomials are optimal D. V. Bogdanov1 · T. M. Sadykov1 Received: 15 December 2017 / Accepted: 13 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract With any integer convex polytope P ⊂ Rn we associate a multivariate hypergeometric polynomial whose set of exponents is Zn ∩ P. This polynomial is defined uniquely up to a constant multiple and satisfies a holonomic system of partial differential equations of Horn’s type. We prove that under certain nondegeneracy conditions any such polynomial is optimal in the sense of [7], i.e., that the topology of its amoeba [13] is as complex as it could possibly be. Using this, we derive optimal properties of several classical families of multivariate hypergeometric polynomials.
1 Introduction Zeros of hypergeometric functions are known to exhibit highly complicated behavior. The univariate case has been extensively studied both classically (see, e.g., [12,15]) and recently (see [3,4,24] and the references therein). Already the distribution of zeros of polynomial instances of the simplest non-elementary hypergeometric function 2 F1 (a, b; c; x) is far from being clear. When one of the parameters a, b equals a nonpositive integer, say a = −d, the series representing 2 F1 terminates and the hypergeometric function is a polynomial of degree d in x (see [3]). By letting the parameters a, b, c assume values in various ranges, one can obtain a wide variety of shapes. Some of them are highly regular (see, e.g., Fig. 1) while other are nearly chaotic. In the present paper, we introduce a definition of a multivariate hypergeometric polynomial in n ≥ 2 complex variables that is coherent with the properties of classical hypergeometric polynomials. This polynomial is defined by an integer convex polytope P ⊂ Rn , its set of exponents is Zn ∩ P. For this polynomial to be “truly hypergeometric” in the sense made precise below, we need to assume that any pair of points in Zn ∩ P can be connected by a polygonal line with unit sides and integer vertices. This assumption does not affect the
This research was performed in the framework of the basic part of the scientific research state task in the field of scientific activity of the Ministry of Education and Science of the Russian Federation, Grant no. 2.9577.2017/8.9.
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T. M. Sadykov [email protected] Laboratory of Artificial Intelligence, Plekhanov Russian University of Economics, 115054 Moscow, Russia
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D. V. Bogdanov, T. M. Sadykov Fig. 1 The hypergeometric aster: zeros of the polynomials 2F1 (−12, b; c; x) with b, c∈ k 1000 : k = 100, . . . , 4000
generality of the results since any polytope that does not satisfy this condition gives rise to a finite number of hypergeometric polynomials that can be considered independently. The following definition is central in the paper and brings together the intrinsic properties of the classical families of hypergeometric polynomials: the denseness, convexity, and irreducibility of the support, as well as the property
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