Compatible Polynomial Interpolation on Algebraic Varieties

Definitions and mathematical statement

Let $V_1,\ldots,V_m\subseteq{\mathbb{C}}^d$ be given algebraic varieties, and let $p_1,\ldots,p_m\in R:={\mathbb{C}}[x_1,\ldots,x_d]$ be given polynomials. For a polynomial $g\in R$, its restriction $g|_{V_j}$ is the function $V_j\to{\mathbb{C}}$ given by $z\mapsto g(z)$.

Determine necessary and sufficient conditions on the data $(V_j,p_j)_{j=1}^m$ for the following existence statement to hold: \[ \exists f\in{\mathbb{C}}[x_1,\ldots,x_d]\quad \forall j\in\{1,\ldots,m\}\quad \forall z\in V_j,\qquad f(z)=p_j(z). \] The requested agreement is pointwise agreement on each variety; the supplied statement imposes no derivative conditions, degree bound, disjointness assumption, or prior compatibility condition on the given polynomials.

The source does not specify whether its use of "algebraic variety" requires irreducibility. This convention remains unresolved, and no irreducibility hypothesis is added here. The supplied literature assessment refers to radicality hypotheses without identifying the ideals or hypotheses involved; that assessment is not adopted as an additional hypothesis or a mathematical conclusion.

Short English statement

Characterize when prescribed polynomial functions on finitely many algebraic varieties in ${\mathbb{C}}^d$ are restrictions of one polynomial on the ambient space. The source's convention concerning irreducibility remains unspecified.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30005011$, title "Compatible Polynomial Interpolation on Algebraic Varieties," and exact source problem code catalogue entry. No individual proposer is identified in the record; its proposed year is $2022$.
• Supplied report citation: Mini-Workshop: Interpolation, Approximation, and Algebra (2023), Oberwolfach Reports, pp. 383--428, DOI 10.4171/owr/2022/7. https://doi.org/10.4171/owr/2022/7. The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The citation's $2023$ year and the record's $2022$ year are retained without reconciliation.
• Supplied literature reference: "McKinley--Shekhtman--Tuesink report results," https://doi.org/10.4171/owr/2022/7. Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that Hilbert--Nullstellensatz and ideal-theoretic criteria solve disjoint varieties and provide exact criteria under radicality hypotheses, while the fully general data-dependent interpolation problem remains only partially characterized. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. This is a supplied verification claim, not verification performed for this draft.