Tchebyshev triangulations of stable simplicial complexes

Abstract

We generalize the notion of the Tchebyshev transform of a graded poset to a triangulation of an arbitrary simplicial complex in such a way that, at the level of the associated F -polynomials j f j −1 ((x − 1)/2) j , the triangulation induces taking the Tchebyshev transform of the first kind. We also present a related multiset of simplicial complexes whose association induces taking the Tchebyshev transform of the second kind. Using the reverse implication of a theorem by Schelin we observe that the Tchebyshev transforms of Schur stable polynomials with real coefficients have interlaced real roots in the interval (−1, 1), and present ways to construct simplicial complexes with Schur stable F -polynomials. We show that the order complex of a Boolean algebra is Schur stable. Using and expanding the recently discovered relation between the derivative polynomials for tangent and secant and the Tchebyshev polynomials we prove that the roots of the corresponding pairs of derivative polynomials are all pure imaginary, of modulus at most one, and interlaced.

Generalized Tchebyshev triangulations

“…Proof. The equivalence of the first two statements may be shown by refining the argument presented in [7,Section 6]. It was noted there that the F -polynomial and the h-polynomial of ∆ are connected by the formula…”

“…Let L be the path with two edges, considered in Examples 3.2 and 3.5. Using Theorem 3.4, as an immediate generalization of [7,Proposition 4.4] we obtain that the polynomials U L,2 n (x) are the ordinary Tchebyshev polynomials of the second kind.…”

“…Let k = 1 and let L be the triangulation of the 1-dimensional simplex obtained by adding the midpoint of the 1-simplex as a new vertex, as in Example 3.2. Certain generalized Tchebyshev triangulations induced by this complex L were considered in [7], where it was shown that the face numbers in these triangulations are independent of the numbering of the vertices. Theorem 3.3 generalizes these results even for this particular choice of L.…”

Enumeration of permutations by number of alternating descents

“…From [8,Theorem 2], we get that the polynomials P n (x) have only real zeros, belong to [−1, 1] and the sequence of zeros of P n (x) separates that of P n+1 (x). From [3,Corollary 8.7], we see that the zeros of the derivative polynomials P n (x) are pure imaginary with multiplicity 1, belong to the line segment [−i, i]. In particular, (1 + x 2 ) P n (x).…”

Delannoy Orthants of Legendre Polytopes

“…This polynomial was shown to be related to certain orthogonal polynomials for the order complexes of some spherical posets in [12] and for a triangulation in [13].…”