Hypergeometric polynomials are optimal. (English) Zbl 1446.35086
Summary: With any integer convex polytope \(P \subset \mathbb{R}^n\) we associate a multivariate hypergeometric polynomial whose set of exponents is \(\mathbb{Z}^n \cap P.\) 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 [M. Forsberg et al., Adv. Math. 151, No. 1, 45–70 (2000; Zbl 1002.32018)], i.e., that the topology of its amoeba [G. Mikhalkin, Ann. Math. (2) 151, No. 1, 309–326 (2000; Zbl 1073.14555)] is as complex as it could possibly be. Using this, we derive optimal properties of several classical families of multivariate hypergeometric polynomials.
MSC:
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
| 47B06 | Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
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