Attainable Zero Counts of Nonnegative Bivariate Polynomials

Definitions and mathematical statement

For a real polynomial $P\in{\mathbb{R}}[x,y]$, let \[ Z_{{\mathbb{R}}}(P)=\{(a,b)\in{\mathbb{R}}^2:P(a,b)=0\}. \] The polynomial is globally nonnegative if $P(a,b)\geq0$ for every $(a,b)\in{\mathbb{R}}^2$. Its real zeros are isolated if, for every $z\in Z_{{\mathbb{R}}}(P)$, there exists $r>0$ such that \[ Z_{{\mathbb{R}}}(P)\cap\{w\in{\mathbb{R}}^2:\|w-z\|_2<r\}=\{z\}. \] Zeros are counted as distinct points, without multiplicity.

For a degree parameter $n$, let $\mathcal P_n$ denote the intended class of globally nonnegative real bivariate polynomials of degree $n$ with isolated real zeros. The supplied record does not specify the degree convention or the admissible values of $n$; these remain unresolved parts of this definition.

The record asserts an upper bound $\#Z_{{\mathbb{R}}}(P)\leq n^2$ and reports examples with at least $\lfloor n^2/4\rfloor$ real zeros. These are supplied source claims, not independently verified bounds. The literal attainability question is: for each admissible $n$, is it true that \[ \forall k\in{\mathbb{Z}},\qquad \left\lfloor\frac{n^2}{4}\right\rfloor\leq k\leq n^2 \quad\Longrightarrow\quad \exists P\in\mathcal P_n\text{ such that }\#Z_{{\mathbb{R}}}(P)=k? \] The interval is read inclusively. A definitive mathematical formulation requires confirmation of the intended polynomial class, degree normalization, admissible degrees, and numerical bounds from the report. In particular, the existence of examples with at least a given number of zeros does not by itself establish attainability of that exact endpoint.

Short English statement

For each intended degree $n$, ask whether every integer from $\lfloor n^2/4\rfloor$ through $n^2$ occurs as the number of distinct isolated real zeros of a globally nonnegative real bivariate polynomial in the intended class. The degree convention, admissible degrees, and stated bounds require confirmation. The supplied record labels the problem open; that status has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002500$, source problem code catalogue entry, titled Attainable Zero Counts of Nonnegative Bivariate Polynomials, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2014 and supplies no individual proposer.
• Dirichlet Series and Function Theory in Polydiscs (2014), Oberwolfach Reports, pp. 335--393, DOI 10.4171/owr/2014/06. Supplied report citation and literature-source link: https://doi.org/10.4171/owr/2014/06. No precise problem page or report excerpt is supplied.
• Supplied literature source labelled "real zeros of nonnegative circuit polynomials": https://doi.org/10.1016/j.jpaa.2020.106602. The record supplies no authors, theorem identifier, or supporting excerpt.
• Literature assessment checked at 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem open and reports that the general zero-count question for the nonnegativity cone remains open while special cones and dimensions are understood. The background explicitly calls this dated literature triage and requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its verification note says that the statement was checked against the official Oberwolfach report. This records a supplied verification claim, not verification performed for this draft.