Fields of Definition of Eutactic Forms

Definitions and mathematical statement

For standard background over ${\mathbb{R}}$, let $n\geq 1$, let $A\in M_n({\mathbb{R}})$ be symmetric and positive definite, and put $q_A(x)=x^{\mathsf T}Ax$. Relative to the lattice ${\mathbb{Z}}^n$, define \[ m(A)=\min_{0\ne v\in{\mathbb{Z}}^n}q_A(v),\qquad S(A)=\{v\in{\mathbb{Z}}^n\setminus\{0\}:q_A(v)=m(A)\}. \] The form is eutactic if there exist real numbers $\lambda_v>0$, one for every $v\in S(A)$, such that \[ A^{-1}=\sum_{v\in S(A)}\lambda_v vv^{\mathsf T}. \] This supplies the usual real lattice meaning of eutaxy; it does not specify a number-field generalization absent from the record.

The supplied question is: can one predict the field of definition of a eutactic form over ${\mathbb{Q}}$, or over another number field $k$? Here a number field means a finite extension of ${\mathbb{Q}}$. The record does not define the intended field of definition or the meaning of eutaxy over $k$. Consequently, a unique precise mathematical assertion cannot be recovered from the supplied wording.

In particular, for a real matrix $A=(a_{ij})$, the coefficient field ${\mathbb{Q}}(a_{ij}:1\leq i,j\leq n)$ and the field generated by the coefficients of the normalized matrix $A/m(A)$ are possible conventions, but the source selects neither. It also does not state whether the intended field concerns the form, its similarity class, or coefficients in a eutaxy identity. For a number-field formulation, the relevant embeddings, positivity conditions, lattice or module, and equivalence relation remain unspecified.

The supplied literature assessment additionally reports the question whether every number field admitting a real embedding can occur. A real embedding is a field embedding $k\hookrightarrow{\mathbb{R}}$. This reported realizability question is retained as a source qualification, not identified with the original prediction question: what must have field of definition $k$, under which convention, and in which dimensions is not specified.

Short English statement

The problem asks how to determine the field of definition associated with a eutactic quadratic form over ${\mathbb{Q}}$ or another number field. The record also reports a realizability question for number fields with a real embedding. Precise formulations require definitions of the intended field, normalization, equivalence, and number-field notion of eutaxy; these are missing.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003269$, title Fields of Definition of Eutactic Forms, exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection, described as extracted workshop discussions containing open and partially solved problems. No individual proposer is supplied.
• Supplied report attribution: Mini-Workshop: Computations in the Cohomology of Arithmetic Groups (2017), Oberwolfach Reports, pp. 2941--2973, DOI 10.4171/owr/2016/52. Supplied link: https://doi.org/10.4171/owr/2016/52. The record separately gives proposed year $2016$; the relationship between this year and the title's $2017$ is unresolved.
• Supplied literature source: Martinet problem survey, https://jamartin.perso.math.cnrs.fr/problat.pdf.
• Supplied literature source: AMS lattice survey, https://www.ams.org/journals/bull/2004-41-04/S0273-0979-04-01018-3/S0273-0979-04-01018-3.pdf.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the field-of-definition question remains open, with surveys asking whether every number field with a real place can occur. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The record expressly calls for independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with a supplied explanation that mathematical content was preserved while notation, typography, or omitted definitions were repaired. This metadata does not establish mathematical correctness or resolve the formulation issues above. No cited source was independently consulted for this draft.