Detecting Nonnormal Smooth Lattice Polytopes

Definitions and mathematical statement

Let $P$ be a lattice polytope in a lattice $M$ and use the induced lattice in its affine span. Write \[ E_P(m)=\#(mP\cap M),\qquad mP=\{mx:x\in P\}, \] for its Ehrhart counting function. A smooth full-dimensional lattice polytope is conventionally one whose primitive edge directions at every vertex form a lattice basis. Normality, or the integer-decomposition property in the usual full-lattice convention, means that for every $m\ge1$, each lattice point of $mP$ is a sum of $m$ lattice points of $P$.

The source requests three exploratory directions:

1. Find examples with $E_P(1)$ substantially smaller than $E_P(2)$, preferably with $P$ smooth. The source gives no ratio, difference, dimension range, or extremal target defining substantially smaller. Smoothness is a preference in this clause, not an unconditional hypothesis for every example.
2. Test smooth polytopes arising from smooth Fano toric varieties and from small projective embeddings in the intended search framework. No dimension bound, list of varieties, line bundles, or measure of smallness is supplied. A smooth Fano variety has ample anticanonical bundle, but the record does not restrict all the associated polytopes to the anticanonical embedding.
3. Determine whether useful invariants, including Ehrhart functions, detect nonnormality. The source does not demand that a chosen invariant give a complete necessary-and-sufficient criterion or specify whether detecting means a sufficient certificate, an effective algorithm, or distinguishing examples.

These tasks are not a single conjecture that a nonnormal smooth polytope exists. A large increase in lattice-point counts, a smooth-polytope search, and a certificate of failure of integer decomposition are distinct issues. The title's nonnormal-smooth wording does not authorize adding a requested counterexample to a general normality statement. The assessment additionally mentions triangulations, but gives no separate triangulation question; no fourth target is invented.

Short English statement

The entry asks for sparse-versus-dilated lattice-point examples, tests in selected smooth toric families, and invariants useful for detecting nonnormality. Smoothness is partly a preference, and the source does not explicitly demand a nonnormal smooth example or a complete Ehrhart criterion.

Sources

• Local UnsolvedMath registry, permanent ID 30000815, code catalogue entry, "Detecting Nonnormal Smooth Lattice Polytopes." Proposed year 2007; proposer unspecified.
• Mini-Workshop: Projective Normality of Smooth Toric Varieties (2008), Oberwolfach Reports, pp. 2283--2320, DOI 10.4171/owr/2007/39; https://doi.org/10.4171/owr/2007/39. The displayed report year 2008 differs from the DOI and proposed year 2007.
• Supplied "normality examples": https://www.normaliz.uni-osnabrueck.de/documentation/interesting-and-challenging-examples-for-normaliz/.
• Statement status corrected_verified; the supplied note says polytope/dilation notation and prose styling were normalized without changing the three directions.
• Literature triage checked 2026-08-21: open. It describes exploratory sparse-dilation, triangulation, and nonnormality-detection questions without a complete resolution. The statement contains no explicit separate triangulation task. Created 2026-08-25; updated 2026-09-16. Exact targets and current status require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and literature provenance remain as supplied. Unreviewed statement-only draft.