Dimension Constraints for Quadratic Forms over Quadric Fields

Definitions and mathematical statement

Let $F$ be a field, with no restriction on its characteristic stated in the supplied record. A quadratic form on a finite-dimensional $F$-vector space $V$ is a map $r:V\to F$ such that $r(cv)=c^2r(v)$ and $r(u+v)-r(u)-r(v)$ is bilinear in $u,v$. Write $\dim(r)=\dim_F(V)$. The form is anisotropic if $r(v)=0$ implies $v=0$. For a field extension $L/F$, write $r_L$ for scalar extension and interpret its isotropy index as \[ i_0(r_L)=\max\{\dim_L U: U\subseteq V\otimes_F L\text{ is an }L\text{-linear subspace and }r_L|_U=0\}. \] This interpretation includes possibly singular forms in characteristic $2$; the source does not explicitly define $i_0$ or impose nonsingularity.

Let $p$ and $q$ be anisotropic quadratic forms over $F$, and let $Q_p\subseteq{\mathbb{P}}(V_p)$ be the projective quadric defined by $p=0$. Whenever $Q_p$ is nonempty and integral, write $F(p)=F(Q_p)$ for its function field. Let $s\in{\mathbb{Z}}$ be the unique integer satisfying \[ 2^s<\dim(p)\leq 2^{s+1}, \qquad k=\dim(q)-2i_0(q_{F(p)}). \] The supplied question asks whether, for every such pair over every field in the intended class, one has \[ k\geq0 \quad\text{and}\quad \exists a\in\mathbb N\ \exists\epsilon\in{\mathbb{Z}}:\qquad -k\leq\epsilon\leq k, \qquad \dim(q)=a\,2^{s+1}+\epsilon. \] The record does not specify whether $0\in\mathbb N$; that convention is left unresolved here because it changes the assertion. It also omits a dimension hypothesis on $p$ ensuring that $F(p)$ exists: a one-dimensional anisotropic form defines an empty projective quadric. Thus the displayed question preserves the supplied target wherever its function-field notation is defined; its unrestricted formulation requires clarification. No characteristic restriction or nonsingularity hypothesis has been added.

Short English statement

The question asks whether the isotropy acquired by $q$ over the function field of the quadric defined by $p$ forces $\dim(q)$ to lie within $k$ of a permitted multiple of $2^{s+1}$, and whether $k$ is always nonnegative. The convention for $\mathbb N$, the scope of quadratic forms, and the missing dimension hypothesis require review. The supplied status is partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30003792$, exact source problem code catalogue entry; title: Dimension Constraints for Quadratic Forms over Quadric Fields. The supplied record gives proposed year $2018$ and names no proposer.
• Supplied report citation: Quadratic Forms and Related Structures over Fields (2019), Oberwolfach Reports, pp. 1235--1291, DOI 10.4171/owr/2018/21. Supplied link: https://doi.org/10.4171/owr/2018/21. The report title, year, pages, and problem attribution have not been independently checked.
• Supplied literature reference, labeled "isotropy indices": https://arxiv.org/abs/1607.07529.
• Supplied literature reference, labeled "quasilinear quadratic forms": https://arxiv.org/abs/1710.09692.
• Supplied literature reference, labeled "extended quasilinear theorems": https://arxiv.org/abs/2409.02059.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and attributes to Scully large cases in characteristic different from $2$ and for quasilinear forms in characteristic $2$, with newer work strengthening the latter. It states that the full dimension-only conjecture remains open. This is supplied dated curation, not an independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied editorial assertions and do not establish mathematical correctness or resolve the qualifications above.