Coefficient Normalization of Positive Definite Homogeneous Forms

Definitions and mathematical statement

For a real homogeneous polynomial $\psi$ of degree $d$, homogeneity means that $\psi(\lambda t)=\lambda^d\psi(t)$ for all $\lambda\in{\mathbb{R}}$ and $t\in{\mathbb{R}}^m$. Positive definiteness is understood as $\psi(t)>0$ for every $t\in{\mathbb{R}}^m\setminus\{0\}$.

The supplied question concerns a linear change of variables and an inequality between coefficients. Its coefficient convention and the allowed linear transformations are not specified. The following is a precise conditional formulation using symmetric ordered-index coefficients and invertible real changes of variables; these conventions require confirmation against the source.

Let $m,d$ be positive integers, and let $\psi\in{\mathbb{R}}[t_1,\ldots,t_m]$ be homogeneous of degree $d$ and positive definite. For a homogeneous form $f$, define its symmetric ordered-index coefficients $n_f(j_1,\ldots,j_d)$ by \[ f(t)=\sum_{j_1=1}^{m}\cdots\sum_{j_d=1}^{m}n_f(j_1,\ldots,j_d)t_{j_1}\cdots t_{j_d}, \] where $n_f(j_{\sigma(1)},\ldots,j_{\sigma(d)})=n_f(j_1,\ldots,j_d)$ for every permutation $\sigma$ of $\{1,\ldots,d\}$. These conditions uniquely determine the coefficients.

Under these conventions, the question is whether, for every such $m,d,\psi$, there exists an invertible real matrix $A\in\operatorname{GL}_m({\mathbb{R}})$ such that the form $\psi'(t)=\psi(At)$ satisfies \[ n_{\psi'}(j_1,\ldots,j_d) \leq \left(\prod_{r=1}^{d}n_{\psi'}(\underbrace{j_r,\ldots,j_r}_{d\text{ indices}})\right)^{1/d} \qquad\text{for every }(j_1,\ldots,j_d)\in\{1,\ldots,m\}^{d}. \] Each diagonal coefficient is $\psi'(e_j)>0$, where $e_j$ is the $j$-th standard basis vector, so the displayed root is the positive real root. The inequality is one-sided, exactly as supplied; no absolute value is imposed on its left-hand side.

Unresolved qualifications. The source does not state whether its coefficients are symmetric ordered-index coefficients, ordinary monomial coefficients, or some other convention. Without a symmetry or indexing convention, an ordered-index expansion does not uniquely determine the individual coefficients. The source also does not specify the group meant by linear equivalence or explicitly state the ranges of $m$ and $d$. Consequently, the conditional formulation above must not be treated as a verified resolution of these omissions.

Short English statement

The supplied problem asks whether every positive definite homogeneous real form can be transformed linearly so that each coefficient is at most the geometric mean of the corresponding diagonal coefficients. A precise conditional version uses symmetric ordered-index coefficients and invertible real changes of variables. The intended coefficient normalization and transformation group remain unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004298$, title: Coefficient Normalization of Positive Definite Homogeneous Forms; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. The record gives proposed year 2019 and names no proposer.
• Supplied report citation: Analytic Number Theory (2020), Oberwolfach Reports, pp. 3141--3205, DOI 10.4171/owr/2019/50. Supplied source URL and literature reference labelled "original OWR report": https://doi.org/10.4171/owr/2019/50. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The report itself has not been independently consulted for this draft.
• Supplied literature assessment, checked exactly 2026-08-22: the record labels the problem open and describes the general coefficient-normalization equivalence as unresolved, with standard reduction results not implying the stated inequality in higher degree and dimension. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.
• The supplied statement status is corrected_verified. Its accompanying qualification says that only formula delimiters, notation, grammar, and typography were repaired, while mathematical content and scope were preserved. This label does not establish mathematical correctness or resolve the formulation issues identified here.