Rationality of Hermite constants

Definitions and mathematical statement

For an integer $n\geq1$, a full-rank lattice in ${\mathbb{R}}^n$ is $\Lambda=A{\mathbb{Z}}^n$ with $A\in\operatorname{GL}_n({\mathbb{R}})$. Its covolume is $\det\Lambda=|\det A|$, and its shortest-vector length is \[ \lambda_1(\Lambda)=\min\{\|v\|_2:v\in\Lambda\setminus\{0\}\}. \] Under the conventional squared-length normalization, the Hermite constant is \[ \gamma_n=\sup_{\Lambda\subset{\mathbb{R}}^n\text{ full-rank lattice}} \frac{\lambda_1(\Lambda)^2}{(\det\Lambda)^{2/n}}. \] It determines the densest lattice packing density of congruent Euclidean balls by $\Delta_n^{\mathrm{lat}}=\operatorname{vol}_n(B(0,1))\gamma_n^{n/2}/2^n$; the density and the Hermite constant are different quantities.

The literal source asks whether $\gamma_n\in{\mathbb{Q}}$ for every $n$, and asks for the arithmetic nature of these constants in dimensions whose exact values are unknown. The supplied summary instead says that Voronoi perfection theory gives $\gamma_n^n\in{\mathbb{Q}}$ for all $n$. Under that supplied assertion, each $\gamma_n$ is algebraic, but rationality of its $n$th power does not imply rationality of $\gamma_n$ itself. The source does not specify whether its rationality question concerns the conventional $\gamma_n$, a power of it, or a differently normalized packing constant. In the conventional normalization some known Hermite constants are irrational, so a universal rationality assertion for $\gamma_n$ cannot be treated as the remaining open problem. A dimension-by-dimension arithmetic classification is a separate target and is not equivalent to determining every exact value.

Short English statement

Determine the arithmetic nature of Hermite constants, distinguishing the constants themselves from their rational powers and from lattice packing densities. The source's universal rationality question depends critically on normalization.

Sources

• Local UnsolvedMath registry, permanent ID 3900007, exact source code catalogue entry, "Rationality of Hermite constants"; proposed by David Eppstein and cited proposers. Registry status partially solved, classification PARTIAL-PROGRESS.
• David Eppstein, The Geometry Junkyard: Open Problems, "Rationality of Hermite constants", https://ics.uci.edu/~eppstein/junkyard/open.html. Supplied HTML extraction accessed July 29, 2026. The index presents the item as open; item-level later resolution was not checked.
• The supplied research summary attributes rationality of $\gamma_n^n$ to Voronoi's perfection theory, and reports exact values known for $n\leq8$ and $n=24$, with other exact values and arithmetic questions unresolved. No precise reference or theorem statement is supplied. The report's exact-value and arithmetic-status claims are retained as supplied assessments, not independent verification or an equivalence of the two tasks.
• Provenance flags status and rights NEEDS_REVIEW: public access was verified, but redistribution permission was not established.