Integral Zeros and Counting for Polynomial Forms

Definitions and mathematical statement

Let $n\geq1$, and let $F\in{\mathbb{Z}}[x_1,\ldots,x_n]$ be a nonzero form, that is, a homogeneous polynomial, of degree $d\geq1$. The first question is to identify conditions under which there exists \[ x\in{\mathbb{Z}}^n\setminus\{0\}\quad\text{with}\quad F(x)=0. \] For real $B\geq1$, define the exact counting function from the source by \[ N_F(B)=\#\{x\in{\mathbb{Z}}^n:F(x)=0,\quad\|x\|_\infty=\max_i|x_i|\leq B\}. \] The second question asks for its behavior as $B\to\infty$, for fixed $F$, particularly for nonsingular cubic and quartic forms beyond the diagonal or sufficiently-many-variable cases treated in the source context.

In the usual projective sense, nonsingularity means that the projective hypersurface $F=0$ over $\overline{{\mathbb{Q}}}$ is smooth, or equivalently that at every nonzero zero of $F$ at least one partial derivative is nonzero. The source does not give a different nonsingularity convention. The count includes the origin, all nonprimitive solutions, both signs, and all scalar multiples within the box; it is not a count of projective points or only primitive solutions. Its nonzero-solution count is $N_F(B)-1$.

This is a program of existence and counting questions. No specific universal main term, variable threshold, local-solubility hypothesis, positivity condition, error exponent, or uniformity as $F$ varies is supplied, so none is added as a conjectured theorem. The desired conditions and asymptotics must depend on an explicitly specified class of forms.

Short English statement

Find conditions guaranteeing a nonzero integral zero of a homogeneous form, and describe the number of all integral zeros in a growing sup-norm box. Nonsingular cubic and quartic forms outside the stated special regimes are the highlighted targets.

Sources

• Local UnsolvedMath registry, permanent ID $30000936$, source code catalogue entry, "Integral Zeros and Counting for Polynomial Forms." No individual proposer is supplied.
• Analytic Number Theory (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/14, pp. 669--746, also listed as "Oberwolfach Report 14/2008": https://doi.org/10.4171/owr/2008/14.
• Supplied reference, "Browning and Heath-Brown, forms in many variables": https://doi.org/10.4171/JEMS/668.
• Supplied literature triage, checked 2026-08-21: status "partially solved." It reports circle-method asymptotics for many nonsingular forms in sufficiently many variables and no known general asymptotic description for arbitrary higher-degree forms. Exact hypotheses and thresholds are not reproduced. This dated assessment derives from supplied OWR curation; time-sensitive status and cited claims require independent verification.