M-Sequence Conditions for Smooth Lattice Polytopes

Definitions and mathematical statement

Let $P\subset{\mathbb{R}}^d$ be a full-dimensional lattice polytope. It is smooth if at each vertex exactly $d$ edges meet and their primitive lattice direction vectors form a basis of ${\mathbb{Z}}^d$. Its Ehrhart $h^*$-vector is defined by \[ \sum_{m\geq0}|mP\cap{\mathbb{Z}}^d|t^m =\frac{h_0^*+h_1^*t+\cdots+h_d^*t^d}{(1-t)^{d+1}}. \] An $M$-sequence is the Hilbert function of a finite-dimensional standard graded commutative algebra: a sequence $(h_0,\ldots,h_d)$ for which some $A=\bigoplus_{i\geq0}A_i$ has $A_0={\mathbb{Q}}$, is generated by $A_1$, satisfies $\dim_{\mathbb{Q}} A_i=h_i$ for $0\leq i\leq d$, and has $A_i=0$ for $i>d$.

The question is whether every smooth lattice polytope $P$, in every dimension, has an $M$-sequence $h^*$-vector. Equivalently, must there exist such a graded algebra $A$ with Hilbert series $\sum_i h_i^*t^i$ for every smooth $P$?

The supplied background says this is a necessary numerical condition for normality. In the usual integer-decomposition convention, normality means that for every integer $m\geq1$, each lattice point of $mP$ is a sum of $m$ lattice points of $P$. The source uses normality only to motivate the weaker numerical question; it does not identify the $M$-sequence condition with normality, unimodular triangulation or unimodular covering. The literature-assessment text concerns triangulation/covering rather than this exact numerical target, so its scope requires checking.

Short English statement

Must the Ehrhart $h^*$-vector of every smooth lattice polytope be the Hilbert function of a standard graded Artinian algebra? This asks for a numerical consequence of normality, without asserting that it is equivalent to normality or a unimodular triangulation.

Sources

• Supplied report: Mini-Workshop: Projective Normality of Smooth Toric Varieties (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/39, pp. 2283--2320; proposed year 2007. https://doi.org/10.4171/owr/2007/39.
• Supplied Egres open-problem summary: https://lemon.cs.elte.hu/egres/open/Integer_decomposition_of_smooth_polytopes.
• Local UnsolvedMath registry, permanent ID 30000818, catalogue entry; created 25 August 2026, updated 16 September 2026. The corrected_verified note says the normality necessary condition and smooth-polytope extension are explicit. Its supplied 21 August 2026 literature triage labels the record open but discusses unimodular triangulation or covering and low-dimensional/special-family results; that is not identical to the displayed $M$-sequence assertion. The dated assessment is retained without treating it as an independently verified status of this exact numerical condition.