Generalized Tameness of an Explicit Polynomial Automorphism

Definitions and mathematical statement

Work in $R={\mathbb{C}}[x,y,z]$ and set \[ \Delta=x^2y+z^2. \] The source specifies an automorphism $\gamma\in\operatorname{Aut}_{{\mathbb{C}}}R$ by \[ \gamma(x)=x,\qquad \gamma(z)=z+x\Delta, \] \[ \gamma(y)=y-2xyz-\Delta^2 +2\left(3z^2\Delta+3xz\Delta^2+x^2\Delta^3\right). \] Every occurrence of $\Delta$ in these defining images is the polynomial $x^2y+z^2$ in the original variables; the images are simultaneous assignments, not successive substitutions.

In the usual polynomial-automorphism terminology, tame automorphisms form the subgroup generated by affine and triangular automorphisms. An exponential automorphism is conventionally $\exp D$ for a locally nilpotent ${\mathbb{C}}$-derivation $D$ of $R$, where for each $a\in R$ some power of $D$ kills $a$ and \[ (\exp D)(a)=\sum_{j\ge0}\frac{D^j(a)}{j!} \] is a finite sum on each polynomial. Let the generalized tame subgroup be generated by tame automorphisms together with these exponential automorphisms.

Is $\gamma$ in that subgroup? This asks for membership by a finite group expression, not merely membership in a closure or stable tameness after adjoining extra variables. The usual locally-nilpotent meaning of exponential should be matched with the report's convention. The statement takes the displayed map's automorphism status as supplied source data; it does not prove membership, nonmembership, or a factorization.

Short English statement

For the exact three-variable automorphism displayed above, is there a finite composition using tame automorphisms and exponentials of locally nilpotent derivations? All coefficients, powers, and simultaneous substitution conventions are retained.

Sources

• Local UnsolvedMath registry, permanent ID 30000661, code catalogue entry, "Generalized Tameness of an Explicit Polynomial Automorphism." Proposed year 2007; proposer unspecified.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, pp. 5--82, DOI 10.4171/owr/2007/01; https://doi.org/10.4171/owr/2007/01. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied note says the polynomial and every automorphism image was enclosed in balanced delimiters, retaining the subgroup-membership question.
• Literature triage checked 2026-08-21: open. It generically calls the extracted item an unresolved problem/conjecture/direction and reports no authoritative complete resolution in the supplied check. No particular theorem or factorization is supplied. Created 2026-08-25; updated 2026-09-16. Current status and exact conventions require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.