Unimodular Coverings from Face and Facet Conditions

Definitions and mathematical statement

Let $P$ be a lattice polytope of dimension $d$ in a lattice $M$. Work relative to its affine lattice $\operatorname{aff}(P)\cap M$, with difference lattice $M_P=\operatorname{span}_{{\mathbb{R}}}(P-P)\cap M$. A full-dimensional simplex $\Delta=\operatorname{conv}(v_0,\ldots,v_d)\subseteq P$, with lattice vertices, is unimodular if $v_1-v_0,\ldots,v_d-v_0$ form a ${\mathbb{Z}}$-basis of $M_P$. A unimodular covering is a finite collection of such simplices satisfying \[ P=\bigcup_{i=1}^N\Delta_i. \] No face-to-face condition on intersections is imposed for a cover. A unimodular triangulation is a triangulation by unimodular simplices; regularity additionally means that the triangulation is induced by a lifting height function. In particular, every regular unimodular triangulation of $P$ gives a unimodular covering of $P$.

Under the usual integer-decomposition convention, normality means \[ \forall m\geq1\ \forall x\in mP\cap M,\qquad x=x_1+\cdots+x_m\text{ for some }x_i\in P\cap M. \] The supplied record names two conditions without defining them: $A(P)$, "pairwise face unimodularity," reported to imply normality, and $B(P)$, "facet unimodularity," reported to imply a regular unimodular triangulation. It asks the two implications \[ A(P)\ \Longrightarrow\ P\text{ has a unimodular covering}\ ? \qquad B(P)\ \Longrightarrow\ P\text{ has a unimodular covering}\ ? \] Here $A$ and $B$ are labels for missing source-specific hypotheses, not definitions recovered from their names.

As literally written, if the stated consequence of $B(P)$ is a regular unimodular triangulation of the whole polytope $P$, its covering consequence is immediate from the definitions. Therefore the second purported open implication cannot be a separate unresolved implication under that reading. Possible differences involving triangulations of facets rather than $P$, or another meaning of the hypothesis, require source review; none is selected here. Normality alone is not substituted for the undefined condition $A(P)$. The workshop title and dated assessment mention smooth polytopes, but the displayed extraction does not explicitly require smoothness.

Short English statement

The record asks whether two face conditions yield a cover by lattice simplices of normalized volume one. The conditions are undefined in the extraction, and the second question is automatic if its stated premise already provides a unimodular triangulation of the whole polytope. This discrepancy prevents a reliable complete reconstruction.

Sources

• Local UnsolvedMath registry, permanent ID $30000817$, exact source code catalogue entry, proposed year 2007; no individual proposer is supplied.
• Mini-Workshop: Projective Normality of Smooth Toric Varieties (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/39, pp. 2283--2320. https://doi.org/10.4171/owr/2007/39.
• Supplied source labelled "covering-properties thesis": https://refubium.fu-berlin.de/bitstream/handle/fub188/26773/Thesis_Codenotti.pdf?sequence=3.
• The supplied corrected_verified note says both sufficient conditions and their shared covering conclusion are explicit in the supplied statement. The assessment checked 2026-08-21 labels the problem open and describes unresolved covering questions and the general smooth-polytope cover problem. This is dated supplied triage; it does not resolve the logical and definitional issues above.