Bounds for Quaternary Local–Global Representation Exceptions

Definitions and mathematical statement

A quaternary quadratic form is a homogeneous polynomial of degree two in four variables. It is positive definite if $Q(x)>0$ for every $x\in{\mathbb{R}}^4\setminus\{0\}$. The supplied question concerns such forms and denotes their discriminant by $D(Q)$, but does not specify the coefficient domain, integrality convention, discriminant normalization, or any primitivity hypothesis.

Under the usual integral-lattice interpretation, let $Q$ be integer-valued on ${\mathbb{Z}}^4$. A positive integer $n$ is represented by $Q$ if there exists $x\in{\mathbb{Z}}^4$ with $Q(x)=n$. It is locally represented if $Q(x_\infty)=n$ for some $x_\infty\in{\mathbb{R}}^4$, and, for every prime $p$, there exists $x_p\in{\mathbb{Z}}_p^4$ with $Q(x_p)=n$, where ${\mathbb{Z}}_p$ is the ring of $p$-adic integers. The real condition is automatic for positive $n$ and positive-definite $Q$. Define \[ E(Q)=\left\{n\in{\mathbb{Z}}_{>0}: n\text{ is locally represented by }Q\text{ and }Q(x)\ne n\text{ for every }x\in{\mathbb{Z}}^4\right\}. \] These definitions describe a conventional interpretation; the supplied record does not explicitly confirm it or impose primitive representation.

The proposed bound, under this interpretation and the uniform reading of $O_\varepsilon$, asks whether \[ \forall\varepsilon>0\ \exists C_\varepsilon>0\ \forall Q\ \forall n\in E(Q),\qquad n\le C_\varepsilon D(Q)^{1+\varepsilon}, \] where $Q$ ranges over the intended class of positive-definite quaternary forms and $C_\varepsilon$ is independent of $Q$ and $n$. The class of forms and the positive discriminant convention must be fixed before this becomes an unqualified statement.

The source phrases the target as a bound for the largest element of $E(Q)$. If $E(Q)$ is nonempty and finite, this is a bound on $\max E(Q)$. The quantified formulation above is vacuous when $E(Q)=\varnothing$, and would imply finiteness for every fixed $Q$; it does not assume that a largest exception already exists. The supplied record gives no convention for the empty case and no separate finiteness hypothesis.

Short English statement

The question asks whether every positive integer represented locally everywhere but not by integer vectors of a positive-definite quadratic form in four variables is bounded by a constant, depending only on $\varepsilon>0$, times $D(Q)^{1+\varepsilon}$. The integral interpretation, exact class of forms, and discriminant convention require confirmation. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30003032$, title: Bounds for Quaternary Local--Global Representation Exceptions. Exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems; proposed year: 2016. No individual proposer is supplied.
• Supplied report citation: Lattices and Applications in Number Theory (2016), Oberwolfach Reports, pp. 87--154, DOI 10.4171/owr/2016/3. https://doi.org/10.4171/owr/2016/3. The background attributes the problem to workshop open-problem discussions documented in these reports; no problem-specific page is supplied.
• Supplied literature source: Rouse, Integers represented by positive-definite quadratic forms. https://arxiv.org/abs/1802.03437.
• Supplied literature source: Oberwolfach Report 3/2016. https://doi.org/10.4171/owr/2016/3.
• Literature assessment checked at 2026-08-21: supplied dated curation, not independently verified. It labels the problem open and states that later work improves bounds for exceptional locally represented integers without establishing the proposed $D(Q)^{1+\varepsilon}$ bound in general. The background expressly calls this dated literature triage and requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its verification note reports only safe notation, typography, and TeX-delimiter cleanup, with hypotheses, quantifiers, and subquestions preserved. This metadata does not constitute independent mathematical verification here.