Real-Rooted Barycentric h-Polynomials and g-Theorem Conditions

Definitions and mathematical statement

Let $P$ be a convex $d$-polytope, not necessarily simplicial. The barycentric subdivision $\Delta=\operatorname{sd}(\partial P)$ is the simplicial complex whose vertices are nonempty proper faces of $P$ and whose simplices are chains of such faces. If $f_{i-1}$ counts its $(i-1)$-simplices, with $f_{-1}=1$, its $h$-polynomial is \[ h_\Delta(t)=\sum_{i=0}^{d}f_{i-1}t^i(1-t)^{d-i}. \] The first conjecture asks whether every complex of this form has $h_\Delta(t)$ with only real zeros.

The second question concerns an arbitrary polynomial \[ h(t)=\sum_{i=0}^{d}h_i t^i\in{\mathbb{Z}}[t],\qquad h_0=1,\quad h_i\geq1, \] with all zeros real, and asks whether its coefficient sequence satisfies the classical $g$-theorem conditions. In their usual full form these require symmetry $h_i=h_{d-i}$ and that \[ g=(1,h_1-h_0,\ldots,h_{\lfloor d/2\rfloor}-h_{\lfloor d/2\rfloor-1}) \] be an $M$-sequence, equivalently a Hilbert function of a standard graded Artinian algebra. In Macaulay form, its entries are nonnegative and obey $g_{i+1}\leq g_i^{\langle i\rangle}$: if $a=\binom{a_i}{i}+\cdots+\binom{a_j}{j}$ is the usual strictly decreasing binomial expansion, then $a^{\langle i\rangle}=\binom{a_i+1}{i+1}+\cdots+\binom{a_j+1}{j+1}$.

The general polynomial hypotheses do not include symmetry, and do not force it: $1+3t+2t^2=(1+t)(1+2t)$ satisfies them but has $h_2\ne h_0$. Hence a literal implication to the full classical $g$-theorem conditions needs a missing symmetry hypothesis or a weaker intended meaning of those conditions. The original record mentions unimodality and Macaulay first-difference inequalities without stating symmetry. Both questions and that discrepancy are retained; symmetry is not silently added, nor is the second question declared equivalent to the first.

Short English statement

Ask for real-rootedness of barycentric boundary $h$-polynomials of all convex polytopes. Separately, ask which $g$-theorem inequalities follow from positive integral real-rooted coefficients; the full classical conditions also require symmetry, absent from the supplied general question.

Sources

• Supplied report: Combinatorics (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/01, pp. 5--78; proposed year 2008. https://doi.org/10.4171/owr/2008/01.
• Supplied Brenti--Welker reference on barycentric subdivisions: https://arxiv.org/abs/math/0606356.
• Supplied Athanasiadis reference on local $h$-polynomials: https://arxiv.org/abs/2402.06219.
• Local UnsolvedMath registry, permanent ID 30000903, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note reports comparison with the official report. The supplied 21 August 2026 triage labels the combined record partially solved, reporting simplicial/Cohen--Macaulay and selected shellable cases while retaining the general nonsimplicial-polytope question. The original separately describes the second conjecture using unimodality and Macaulay inequalities. This dated assessment is not independently updated here.