Fixed-length sums of 2/(x^2-2)

Definitions and mathematical statement

For a field $F$ of characteristic zero, let \[ V_n(F)=\left\{\sum_{i=1}^n\frac{2}{x_i^2-2}:x_i\in F,\ x_i^2\ne2\right\},\qquad n\geq1. \] Poonen's AIM question asks to characterize $V_n({\mathbb{Q}})$ for fixed $n$. It states the necessary integrality conditions $|\alpha|_p\leq1$ at $p=2$ and at every odd prime with Legendre symbol $(2/p)=-1$, and asks whether these suffice. The formula is reconstructed from damaged fraction/superscript formatting; the source explicitly takes finite $x_i\in{\mathbb{Q}}$, not points at infinity.

The report claims for every odd prime with $(2/p)=-1$ that \[ V_2({\mathbb{Q}}_p)={\mathbb{Z}}_p. \] At the ramified prime it gives only the following exact residue images: \[ V_1({\mathbb{Q}}_2)\bmod8=\{0,1,6,7\},\qquad V_2({\mathbb{Q}}_2)\bmod8=\{0,1,2,4,5,6,7\}. \] Thus $\alpha=3$ satisfies all printed integrality conditions but is not in $V_2({\mathbb{Q}}_2)$ or $V_2({\mathbb{Q}})$. In contrast $3\in V_3({\mathbb{Q}})$, for instance all three variables can equal $2$. This refutes bare integrality sufficiency for $n=2$, not the Hasse principle, since the counterexample already fails local solubility. The residue assertion does not classify every $2$-adic value in each allowed residue class.

The report also gives an exact one-summand criterion, which in this convention is \[ \alpha\in V_1({\mathbb{Q}})\quad\Longleftrightarrow\quad \alpha\ne0\ \text{and}\ 2+2/\alpha\in{\mathbb{Q}}^2. \] Its norm-one-torus reformulation uses $L={\mathbb{Q}}(\sqrt2)$ and $u=(x+\sqrt2)/(x-\sqrt2)$, with $N_{L/{\mathbb{Q}}}(u)=1$ and $u\ne1$. The trace identity yields \[ \alpha\in V_n({\mathbb{Q}})\quad\Longleftrightarrow\quad \exists u_1,\ldots,u_n\in L^\times:\ N(u_i)=1,\ u_i\ne1, \quad\sum_i\operatorname{Tr}_{L/{\mathbb{Q}}}(u_i)=4\alpha+2n. \] Zero-sum padding gives the reported length inclusions; explicitly $1+(-1)=0$ and $1+1+(-2)=0$ are values at $(2,0)$ and $(2,2,1)$, so $V_n\subset V_{n+2}$ and $V_n\subset V_{n+3}$. No general fixed-length global classification follows from these reductions.

The source remarks suggest analogous Waring-type questions for rational functions without rational poles, note unspecified earlier work when poles are rational, and propose the local-global formulation (Swinnerton-Dyer). They suggest local solubility for sufficiently large fixed $n$ and a Diophantine-definition application if the full integrality characterization holds: integrality at the inert half of the primes, together with $2$. Those are motivations, not established consequences of the two-summand result.

Short English statement

Characterize fixed-length sums of $2/(x^2-2)$ over ${\mathbb{Q}}$. The report shows that the stated integrality conditions miss a genuine $2$-adic obstruction for two summands: $3$ is excluded, although three summands represent it. It also supplies a one-term test, a norm-torus trace reduction and padding identities, leaving the full local-global question separate.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent ID $20000448$, source code catalogue entry; original question, every remark and supplied research summary.
• AIM workshop "Rational and integral points on higher dimensional varieties", Question 15, attributed to Poonen: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf. Canonical location aim-algebraic-number-theory-notes.json, notes[99]; original literature field empty. No PDF was opened here.
• Original remarks attribute the Hasse-principle reformulation to Swinnerton-Dyer and the rational-function/Diophantine motivations to Poonen; no references for the mentioned previous work are supplied.
• Supplied machine-generated research attempt 1: partial result, problem open at the run and currently partially solved; rigor metadata "proved", novelty confidence low. The non-peer-reviewed local and torus results require independent verification before citation.