On Relations for Zeros of f-Polynomials and f +-Polynomials
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Let Φ be an irreducible (possibly noncrystallographic) root system of rank l of type P. For the corresponding cluster complex Δ(P), which is known as pure (l − 1)-dimensional simplicial complex, we define the generating function of the number of faces of Δ(P) with dimension i − 1, which is called f-polynomial. We show that the f-polynomial has exactly l simple real zeros on the interval (0, 1) and the smallest root for the infinite series of type A l , B l , and D l monotone decreasingly converges to zero as the rank l tends to infinity. We also consider the generating function (called the f +-polynomial) of the number of faces of the positive part Δ+(P) of the complex Δ(P) with dimension i − 1, whose zeros are real and simple and are located in the interval (0, 1), including a simple root at t = 1. We show that the roots in decreasing order of f-polynomial alternate with the roots in decreasing order of f +-polynomial.
KeywordsGrowth function The Jacobi polynomials
Mathematics Subject Classification (2010)12D10
The author was very glad to participate in the symposium “the 3rd Franco - Japanese - Vietnamese Symposium on Singularities” Hanoi, Vietnam, November 30–December 4, 2015. The author thanks all the organizers. The author is grateful to Kyoji Saito for enlightening discussions and his great encouragement. The author is grateful to Mutsuo Oka for his warm encouragement. This researsh was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
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