Zero Thresholds for Positive Semidefinite Ternary Forms

Definitions and mathematical statement

Under the intended real interpretation, a ternary form of degree $2k$ is a homogeneous polynomial $p\in{\mathbb{R}}[x,y,z]_{2k}$, where ${\mathbb{R}}[x,y,z]_m$ denotes the space of homogeneous polynomials of degree $m$, including the zero polynomial. It is positive semidefinite if $p(v)\geq0$ for every $v\in{\mathbb{R}}^3$. Define its set of distinct real projective zeros by \[ Z_{{\mathbb{R}}}(p)=\{[v]\in\mathbb{RP}^{2}:p(v)=0\}, \] where $[v]$ is the equivalence class of a nonzero real vector under multiplication by nonzero real scalars. An indefinite real form $h$ means a homogeneous polynomial taking both positive and negative values on ${\mathbb{R}}^3$.

For each integer $k\geq3$, let $\alpha(k)$ be the least integer $a$, provided such a least integer exists, with the following property: for every $p\in{\mathbb{R}}[x,y,z]_{2k}$, \[ \left(p(v)\geq0\text{ for all }v\in{\mathbb{R}}^3\ \text{ and }\ \#Z_{{\mathbb{R}}}(p)>a\right) \Longrightarrow \left\{ \begin{gathered} \text{there exist an integer }d\text{ with }1\leq d\leq k,\\ h\in{\mathbb{R}}[x,y,z]_d\text{ indefinite, and }q\in{\mathbb{R}}[x,y,z]_{2k-2d},\\ p=h^2q. \end{gathered} \right. \] The zero polynomial is included because the supplied statement does not exclude it. An infinite projective zero set satisfies the strict inequality against every finite integer $a$.

The supplied conjecture is \[ \alpha(k)\geq k^2+1\qquad\text{for every integer }k\geq3. \] The source record does not explicitly specify the coefficient field, whether projective zeros are real and counted distinctly, or the precise convention for indefiniteness. The definitions above state the natural real, sign-changing interpretation; confirmation against the source is required. The existence of a finite least threshold is presupposed by the supplied formulation and is not established here.

Short English statement

For each integer $k\geq3$, consider the least zero-count threshold above which every nonnegative real ternary form of degree $2k$ has a square factor whose underlying form changes sign. The conjecture says that this threshold is at least $k^2+1$. The real-projective interpretation and the existence of the threshold require source confirmation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002562$, title: Zero Thresholds for Positive Semidefinite Ternary Forms; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is 2014.
• Supplied source citation: Real Algebraic Geometry With A View Toward Systems Control and Free Positivity (2015), Oberwolfach Reports, pp. 977--1045, DOI 10.4171/owr/2014/17. Supplied source and literature reference, labeled "Oberwolfach Report 2014/17": https://doi.org/10.4171/owr/2014/17. The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Literature assessment checked on 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem open, states that the source presents the inequality as a conjecture, and reports that all but five values were proved. It says that no authoritative proof of the remaining cases was identified and consequently classifies the conjecture as open. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review of the cleanup. These are supplied editorial assertions, not independent mathematical verification of this draft.