Equivalence of Explicit Affine Hypersurfaces

Definitions and mathematical statement

Retain the source's intended polynomials and recurrence over ${\mathbb{C}}$: \[ H_0=Y,\qquad H_1=X,\qquad r=X^3+XYZ-Y^3,\qquad a_n=\deg H_n, \] \[ H_{n+1}=\frac{r^{a_n}+H_n^3}{H_{n-1}}\qquad(n\geq1). \] For a polynomial $P$, write $V(P)=\{z\in{\mathbb{C}}^3:P(z)=0\}$. The proposed hypersurfaces are \[ U_n=V(H_n-1),\qquad V_n=V(XY-Z^{a_n}-1). \] For every $n\geq3$, the source asks whether there exists a holomorphic automorphism $\Phi_n:{\mathbb{C}}^3\to{\mathbb{C}}^3$ with $\Phi_n(U_n)=V_n$, and separately whether such ambient equivalence holds by a real diffeomorphism or by a homeomorphism. In the usual underlying-real-manifold interpretation these latter maps act on ${\mathbb{C}}^3\cong{\mathbb{R}}^6$; equivalence only of real loci in ${\mathbb{R}}^3$ would be a different question.

The recurrence as printed does not define a polynomial sequence. Since $a_1=1$, its first step gives \[ H_2=\frac{r+X^3}{Y}=\frac{2X^3+XYZ-Y^3}{Y}. \] The numerator restricts to $2X^3$ at $Y=0$, so it is not divisible by $Y$ over ${\mathbb{C}}$. Consequently $H_2$ is not a polynomial, and the intended polynomial degree $a_2$ and subsequent hypersurfaces are not determined in the stated sense. A rational-function degree or a numerator-clearing convention would change the construction and is not supplied. No sign, exponent or definition is silently corrected; the three ambient-equivalence questions are retained pending recovery of the intended recurrence.

Short English statement

Compare the specified hypersurfaces by holomorphic, smooth real and topological ambient changes of coordinates. The printed recurrence fails to produce a polynomial already at $H_2$, so the hypersurface family must be corrected from the source before those questions are fully defined.

Sources

• Supplied report: Affine Algebraic Geometry (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/01, pp. 5--82; proposed year 2007. https://doi.org/10.4171/owr/2007/01.
• Local UnsolvedMath registry, permanent ID 30000658, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note reports balancing recurrence, degree and hypersurface notation while retaining equivalence questions; it does not explain divisibility of the displayed recurrence. The supplied 21 August 2026 literature triage labels the extracted claim open and reports no located authoritative complete resolution. No additional external reference is supplied; no independent status assessment is asserted.