Uniform Sum-of-Squares Multipliers on Real Projective Curves

Definitions and mathematical statement

Let $X\subseteq{\mathbb{P}}^n_{{\mathbb{R}}}$ be an irreducible nondegenerate real projective curve. Here nondegenerate means that $X$ is not contained in a hyperplane, and we interpret the supplied requirement of dense real points as Zariski density of $X({\mathbb{R}})$ in $X$. Write \[ A={\mathbb{R}}[x_0,\ldots,x_n]/I(X),\qquad A_d=\text{the homogeneous component of degree }d. \] All polynomial identities below are identities in $A$, so sums of squares are taken modulo the homogeneous vanishing ideal $I(X)$. For an integer $d\geq0$, set \[ \Sigma_{X,2d}=\left\{\sum_{i=1}^{r}h_i^2:r\in{\mathbb{Z}}_{\geq0},\ h_i\in A_d\right\}. \] Define $P_{X,2d}\subseteq A_{2d}$ to consist of those forms whose values are nonnegative at every nonzero real homogeneous representative of a point of $X({\mathbb{R}})$. This condition is independent of the representative because the degree is even.

Requested bound. Given an even degree $2s$, interpreted as $s\in{\mathbb{Z}}_{\geq0}$, find an explicit integer $k\geq0$, depending only on $s$ and invariants of $X$, such that \[ \forall p\in P_{X,2s}\quad \exists q\in\Sigma_{X,2k}\quad pq\in\Sigma_{X,2(s+k)}. \] Thus the same $k$ must work for every such $p$; the multiplier $q$ may depend on $p$. The source does not specify which invariants are to control $k$.

Unresolved qualifications. The supplied wording does not require $q\neq0$. A nontrivial multiplier formulation would require $q\in\Sigma_{X,2k}\setminus\{0\}$, but that additional requirement needs confirmation and has not been silently inserted into the displayed statement. The supplied statements concern curves, whereas the background literature assessment describes a general variety-level question. No precise higher-dimensional formulation or its hypotheses are supplied; the displayed statement therefore records the curve formulation only. No smoothness assumption is supplied.

Short English statement

For an irreducible nondegenerate real projective curve with Zariski-dense real points, seek an explicit degree bound, uniform over all nonnegative forms of a fixed even degree, for a sum-of-squares multiplier making their product a sum of squares on the curve. The permitted invariants, the intended nonzero-multiplier condition, and the background's broader variety-level target remain unresolved.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30002566$, source problem code catalogue entry, title "Uniform Sum-of-Squares Multipliers on Real Projective Curves." The record gives proposed year 2014 and supplies no named proposer.
• Supplied report 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. https://doi.org/10.4171/owr/2014/17. The supplied literature list labels this "Oberwolfach Report 2014/17."
• Supplied literature source: Blekherman--Smith--Velasco, Journal de Mathematiques Pures et Appliquees. https://doi.org/10.1016/j.matpur.2018.12.010.
• Supplied literature source: Blekherman--Smith--Velasco, "projective Positivstellensatz." https://arxiv.org/abs/1104.1772.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and states that sharp multiplier-degree bounds are known for nondegenerate totally-real projective curves, while a general variety-level bound remains only partially solved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Its relation to the curve-only statement requires review.