Face numbers of uniform triangulations of simplicial complexes

Abstract

A triangulation of a simplicial complex ∆ is said to be uniform if the fvector of its restriction to a face of ∆ depends only on the dimension of that face. This paper proves that the entries of the h-vector of a uniform triangulation of ∆ can be expressed as nonnegative integer linear combinations of those of the h-vector of ∆, where the coefficients do not depend on ∆. Furthermore, it provides information about these coefficients, including formulas, recurrence relations and various interpretations, and gives a criterion for the h-polynomial of a uniform triangulation to be real-rooted. These results unify and generalize several results in the literature about special types of triangulations, such as barycentric, edgewise and interval subdivisions.

Face numbers of barycentric subdivisions of cubical complexes

“…The same argument, similar to that in the proof of[4, Corollary 5.6], works for k ∈ {0, n + 1}. Part (f) follows from part (d) by straightforward induction on k (for fixed n), where the base k = 0 of the induction holds because of Equation (3).…”

“…b. The main result of [7] implies (see [4,Section 8]) that h(sd(∆), x) has a nonnegative realrooted symmetric decomposition with respect to n for every triangulation ∆ of an n-dimensional ball. Does this hold if ∆ is replaced by any cubical subdivision of the n-dimensional ball?…”

“…The proof is motivated by the one of the aforementioned result of [10], given and extended to the setting of uniform triangulations of simplicial complexes in [4]. To explain, we let h(∆, x) = n+1 i=0 h i (∆)x i denote the h-polynomial and sd(∆) denote the barycentric subdivision of an n-dimensional simplicial complex ∆.…”

Symmetric decompositions, triangulations and real-rootedness

“…Once again, the real-rootedness of (2) has been much less studied. This can be deduced from part (b) of Theorem 1.1, stated in the sequel, when ∆ is the barycentric subdivision of a simplicial ball (see [9,Section 8]), but should be expected to hold in much more general situations. For instance, it seems natural to ask, in the spirit of [19,Question 1], whether it holds for all barycentric subdivisions of polyhedral balls.…”

“…Since Theorem 1.1 is closely related to barycentric subdivision, it is natural to wonder whether there is a similar result which applies to more general types of triangulations. Indeed, part (a) of the theorem was generalized in [9] in the framework of uniform triangulations of simplicial complexes, of which barycentric subdivision is a prototypical example. The operator D n is replaced there by an operator D F ,n : R n [x] → R n [x] which depends on a triangular array of numbers F and maps h(∆, x) to the h-polynomial of a triangulation of ∆ for every (n − 1)-dimensional simplicial complex ∆, provided that the f -vector (prescribed by F ) of the restriction of this triangulation to a face of ∆ depends only on the dimension of that face (the collection of f -vectors of these restrictions is precisely the information encoded in F ).…”