Non-Integrality Conjecture for Rational Hypersurfaces

Definitions and mathematical statement

Let $n\in{\mathbb{Z}}$ with $n\geq 2$, and let ${\mathbb{A}}^n_{{\mathbb{Q}}}$ denote affine $n$-space over ${\mathbb{Q}}$, with coordinate functions $x_1,\ldots,x_n$. Let $V\subseteq{\mathbb{A}}^n_{{\mathbb{Q}}}$ be a hypersurface defined over ${\mathbb{Q}}$, and write $V({\mathbb{Q}})$ for its rational points. A subset of $V({\mathbb{Q}})$ is Zariski dense in $V$ if its Zariski closure in $V$ equals $V$. Define \[ S_V=\{P\in V({\mathbb{Q}}):x_1(P)\notin{\mathbb{Z}}\} =V({\mathbb{Q}})\cap\left(({\mathbb{Q}}\setminus{\mathbb{Z}})\times{\mathbb{Q}}^{n-1}\right). \] Non-Integrality Conjecture, as recorded. Is it true that, for every such $n$ and every such $V$, \[ \overline{V({\mathbb{Q}})}^{\,\mathrm{Zar}}=V \quad\Longrightarrow\quad \overline{S_V}^{\,\mathrm{Zar}}=V? \] Unresolved formulation qualifications. The supplied record does not specify whether the hypersurface must be irreducible or geometrically irreducible. It also imposes no nonconstancy or dominance condition on the restriction $x_1|_V$, and gives no exclusion for hypersurfaces contained in a coordinate hyperplane $x_1=m$ with $m\in{\mathbb{Z}}$. These cases require clarification before the displayed unrestricted question can be treated as a precise statement of the intended open conjecture; no additional hypothesis has been inserted here.

The supplied report commentary attributes the case $n=2$ to Siegel's theorem on integral points on curves. This attribution, including its applicability to the exact hypotheses above, requires source verification and is not asserted here as an independently checked result.

Short English statement

The recorded question asks whether rational points whose first coordinate is not an integer are Zariski dense on every rationally densely populated hypersurface in affine space of dimension at least two. The intended hypotheses need clarification, especially when the first coordinate is constant and integral. The supplied open status and attribution of the plane case to Siegel are unverified source claims.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001146$, source problem code catalogue entry, title Non-Integrality Conjecture for Rational Hypersurfaces, in the collection Oberwolfach Reports Open Problems. The record gives the proposed year as 2009 and names no individual proposer.
• Supplied source citation: The Arithmetic of Fields (2009), Oberwolfach Reports, pp. 285--354, DOI 10.4171/owr/2009/05. Source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2009/05.
• Supplied literature-source link labelled "report statement of NIC": https://ems.press/content/serial-article-files/46207.
• Supplied dated literature curation, checked 2026-08-21: the record labels the problem open, describes it as the Non-Integrality Conjecture for $n\geq2$, and attributes the case $n=2$ to Siegel while describing the general assertion as open. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that the conjecture is complete and that report commentary and affine-space notation were normalized. These are supplied registry qualifications, not verification performed for this draft.