Abundance of Zeros in Families of Quadratic Forms

Definitions and mathematical statement

The supplied record fixes integral quadratic polynomials $Q_1,\ldots,Q_h\in{\mathbb{Z}}[Y_1,\ldots,Y_t]$. Here $t$ and $h$ denote the numbers of variables and polynomials, respectively. Their intended admissible ranges are not specified; the formulation below takes $t\geq1$ and $h\geq2$, since the displayed endpoint is undefined for $h=1$. The source does not specify whether the polynomials must be homogeneous, nor any rank, independence, nonsingularity, or solubility hypotheses.

For a real parameter $B\geq1$ and a fixed real exponent $\theta$, define \[ X_\theta(B)=\{\mathbf x\in{\mathbb{Z}}^h: |x_i|\leq B^\theta\text{ for every }1\leq i\leq h\} \] and, for $\mathbf x\in X_\theta(B)$, define \[ N_{\mathbf x}(B)=\#\left\{\mathbf y\in{\mathbb{Z}}^t: |y_j|\leq B\text{ for every }1\leq j\leq t,\quad \sum_{i=1}^h x_iQ_i(\mathbf y)=0\right\}. \] The requested estimate, under the usual uniform asymptotic interpretation of positive proportion and $\gg$, asks whether there exist a real number $\theta$, constants $c>0$, $0<\delta\leq1$, and $B_0\geq1$, such that \[ \frac{h-7}{h-1}\leq\theta<1 \] and, for every real $B\geq B_0$, \[ \#\{\mathbf x\in X_\theta(B): N_{\mathbf x}(B)\geq cB^{t-2-\theta}\} \geq \delta\,\#X_\theta(B). \] The exponent and constants are independent of $B$; they may depend on the fixed polynomial family, with the constants also allowed to depend on $\theta$. This interpretation counts all integer coefficient vectors in the displayed box, including the zero vector, because no exclusions are supplied.

This is a statement-only reconstruction of the supplied question, with unresolved scope. The record does not establish whether the question is intended for every fixed family or only families satisfying omitted hypotheses. It does not specify whether positive proportion is measured in the entire box or within an admissible subfamily. No additional restriction $\theta\geq0$ is supplied: when $2\leq h\leq6$, the stated interval permits negative exponents, for which $X_\theta(B)=\{\mathbf0\}$ whenever $B>1$. These qualifications require source review before the intended assertion can be fixed.

Short English statement

For a fixed family of integral quadratic polynomials, ask whether some exponent $(h-7)/(h-1)\leq\theta<1$ makes a positive proportion of integer coefficient vectors of size at most $B^\theta$ yield at least $cB^{t-2-\theta}$ integer zeros in the box of size $B$, uniformly for sufficiently large $B$. The permitted polynomial families, parameter ranges, and population used to measure proportion remain unclear.

Sources

• Local UnsolvedMath registry, permanent ID $30003281$, source problem code catalogue entry, title "Abundance of Zeros in Families of Quadratic Forms," in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2016; no individual proposer is identified.
• Supplied source citation: Analytic Number Theory (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/53, pp. 2975--3030. Supplied source URL: https://doi.org/10.4171/owr/2016/53. The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature reference: "Oberwolfach Report 53 (2016)," https://doi.org/10.4171/owr/2016/53. Literature checked at: 2026-08-21. This is supplied dated curation, not an independently verified assessment. It labels the problem open and reports that no authoritative proof or disproof of this exact positive-proportion estimate was found. The supplied background requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. These are supplied editorial claims, not independent mathematical verification of this draft.