Thin-deleted asymptotic and an explicit split-fibre surplus

Definitions and mathematical statement

In the rational-point interpretation of the supplied report, let \[ V=\left\{([x_0:\cdots:x_3],[y_0:\cdots:y_3])\in{\mathbb{P}}^3_{{\mathbb{Q}}}\times{\mathbb{P}}^3_{{\mathbb{Q}}}: \sum_{i=0}^3x_i y_i^2=0\right\}, \qquad U=V\cap\{x_i y_i\ne0\text{ for every }i\}. \] Choose primitive integral coordinate vectors $x,y$ separately for the two projective factors. Define \[ H(x,y)=\left(\max_i|x_i|\right)^3\left(\max_i|y_i|\right)^2, \qquad N_U(B)=\#\{P\in U({\mathbb{Q}}):H(P)\le B\}\quad(B\ge1). \] This is independent of the remaining sign choices; each projective pair counts once. Peyre's question is to estimate this counting function on the locus obtained by deleting every point with any zero $x_i$ or $y_i$. The original sentence does not explicitly state the ground field or primitive-coordinate convention; these are made explicit as the rational-height reading used by the report.

The supplied later assessment distinguishes this literal coordinate-open count from a further thin-deleted count. It reports that Browning and Heath-Brown's known Peyre asymptotic transfers to the exact coordinate-open locus after the necessary thin deletion, with the residual coordinate-boundary contribution only $O(B)$. Writing $T$ for that additional thin exceptional set, its reported form is \[ N_{U\setminus T}(B)\sim c_V B\log B. \] Here $c_V$ denotes the report's Peyre constant (called $c_X$ there). The exact set $T$, the full theorem hypotheses and a formula for the constant are not supplied; this notation records the reported result without choosing a new deletion.

On the literal $U$, the report's explicit split fiber lies over \[ x^*=[1:1:-1:-1],\qquad y_0^2+y_1^2-y_2^2-y_3^2=0. \] Let $N_{x^*}(B)$ count its points with all $y_i\ne0$ and the same height bound. The candidate proposition gives \[ N_{x^*}(B)\ge\zeta(2)^{-2}B\log B-O(B),\qquad \zeta(2)=\sum_{m\ge1}m^{-2}, \] and reports the combined consequence \[ \liminf_{B\to\infty}\frac{N_U(B)}{B\log B} \ge c_V+\zeta(2)^{-2}. \] Thus the source's literal coordinate deletion cannot simply be identified with the thin-deleted asymptotic having coefficient $c_V$. The report explicitly leaves a full asymptotic incorporating all split fibers unproved. These are supplied partial-result claims, not a new proof or an exact leading coefficient for $N_U$.

Short English statement

Estimate rational points on $\sum x_i y_i^2=0$ with the stated height and every coordinate nonzero. The supplied report distinguishes a known asymptotic requiring an additional thin deletion from the literal source count, where one explicit split fiber already contributes a further $B\log B$ term.

Sources

• Local UnsolvedMath registry, permanent ID $20000436$, source code catalogue entry; AIM algebraic number theory corpus, aim-algebraic-number-theory-notes.json, notes[87]. Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0.
• AIM Rational and integral points on higher dimensional varieties, Question 4, attributed to Peyre; workshop December 11--20, 2002, document version November 22, 2004, as supplied in the verification: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf. The source remarks ask for news of ongoing work and end with the fragment "Problem/". The verification reports separated OCR summation indices; no new PDF inspection is claimed.
• The research summary names Browning and Heath-Brown's Peyre asymptotic without a complete reference, exceptional-set definition or constant formula. The original literature field is empty.
• Registry dated August 14, 2026: statement exact, status partially_solved, classification PARTIAL-PROGRESS; attempt 1 labelled partial result, rigor proved, novelty confidence low. The boundary estimate, split-fiber surplus and literature assessment are machine-generated research aids, not peer reviewed, requiring independent verification before citation.