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THE LOCAL $h$-POLYNOMIALS OF CLUSTER SUBDIVISIONS HAVE ONLY REAL ZEROS
Part of
PHILIP B. ZHANG
Journal:

Bulletin of the Australian Mathematical Society / Volume 98 / Issue 2 / October 2018

Published online by Cambridge University Press:

01 August 2018, pp. 258-264

Print publication:

October 2018
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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.