The Real-Zero Polynomial Amalgamation Conjecture

Definitions and mathematical statement

For an integer $s\geq 1$, use the following at-origin convention: a polynomial $f\in{\mathbb{R}}[u_1,\ldots,u_s]$ is real-zero if $f(0)\ne 0$ and, for every $v\in{\mathbb{R}}^s$, every complex root of the univariate polynomial $t\mapsto f(tv)$ is real. Nonzero constant polynomials satisfy this condition. Write ${\mathbb{N}}_0=\{0,1,2,\ldots\}$, and interpret degree as total degree. The supplied record does not explicitly define real-zero or specify a normalization such as $f(0)=1$; the convention above makes the displayed question precise and requires source confirmation.

Amalgamation question, conditional on the literal scalar-variable reading. Is the following assertion true? For every $d\in{\mathbb{N}}_0$ and every pair of real-zero polynomials \[ p\in{\mathbb{R}}[x,y],\qquad q\in{\mathbb{R}}[x,z],\qquad \deg p\leq d,\quad \deg q\leq d, \] where $x,y,z$ are individual indeterminates, if \[ p(x,0)=q(x,0)\quad\text{as polynomials in }{\mathbb{R}}[x], \] then there exists a real-zero polynomial $r\in{\mathbb{R}}[x,y,z]$ with $\deg r\leq d$ such that the polynomial identities \[ r(x,y,0)=p(x,y),\qquad r(x,0,z)=q(x,z) \] hold?

Unresolved scope qualification. The record does not specify whether $x,y,z$ denote individual variables or tuples of variables. If they denote tuples, the same identities require corresponding zero tuples, and the dimensions and their quantifiers must be specified. The scalar formulation above is therefore an explicit conditional reading, not a verified identification of the intended general conjecture. In particular, the supplied literature assessment refers to a remaining two-shared-variable conjecture, which makes this distinction material.

Short English statement

The question asks whether two real-zero polynomials that agree on their shared variables can be extended to one real-zero polynomial without increasing the prescribed degree bound. The source leaves the scalar-versus-tuple interpretation and the real-zero convention unspecified. Its open status and literature assessment are supplied dated curation, not independently verified conclusions.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30004437$, title The Real-Zero Polynomial Amalgamation Conjecture, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The supplied proposed year is 2020; no individual proposer is identified.
• Supplied report citation: Real Algebraic Geometry with a View Toward Hyperbolic Programming and Free Probability (2021), Oberwolfach Reports, pp. 639--712, DOI 10.4171/owr/2020/12. Also listed among the literature sources as Oberwolfach Report 12/2020: https://doi.org/10.4171/owr/2020/12. The record describes these reports as documenting mathematical workshops and their open-problem discussions. No specific page within the cited range is supplied for this problem.
• Supplied literature source: Sawall--Schweighofer (2024), https://doi.org/10.1016/j.indag.2023.08.002. No article title or precise theorem reference is supplied.
• Literature checked at: 2026-08-22. This exact date belongs to supplied dated curation and is not an independent verification. The supplied assessment labels the extracted conjecture open and not settled; it says that Sawall and Schweighofer found counterexamples to the "obvious necessary conditions" and proved only special positive cases, and that their remaining two-shared-variable conjecture does not resolve the general record. The supplied background characterizes this assessment as dated literature triage derived from the OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry supplies statement status corrected_verified, with an explanation that mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired for self-contained publication. This is a supplied editorial designation, not independent mathematical review of this draft.