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THE LOCAL $h$-POLYNOMIALS OF CLUSTER SUBDIVISIONS HAVE ONLY REAL ZEROS

Part of: Polytopes and polyhedra Polynomials, rational functions Combinatorics Algebraic combinatorics

Published online by Cambridge University Press:  01 August 2018

PHILIP B. ZHANG Open the ORCID record for PHILIP B. ZHANG [Opens in a new window]
PHILIP B. ZHANG*
Affiliation:
College of Mathematical Science, Tianjin Normal University, Tianjin 300387, PR China email [email protected]
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Abstract

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Athanasiadis [‘A survey of subdivisions and local $h$-vectors’, in The Mathematical Legacy of Richard P. Stanley (American Mathematical Society, Providence, RI, 2017), 39–51] asked whether the local $h$-polynomials of type $A$ cluster subdivisions have only real zeros. We confirm this conjecture and prove that the local $h$-polynomials for all the Cartan–Killing types have only real roots. Our proofs use multiplier sequences and Chebyshev polynomials of the second kind.

Keywords

real-rootednessmultiplier sequencelocal h-polynomialcluster subdivisionChebyshev polynomial of the second kind

MSC classification

Primary: 26C10: Polynomials
Secondary: 05A15: Exact enumeration problems, generating functions 05E45: Combinatorial aspects of simplicial complexes 52B45: Dissections and valuations (Hilbert's third problem, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 2 , October 2018 , pp. 258 - 264
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was supported by the National Science Foundation of China (Nos. 11626172 and 11701424), the TJNU Funding for Scholars Studying Abroad, the PhD Program of TJNU (No. XB1616) and MECF of Tianjin (No. JW1713).

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