Lifting Polynomial Automorphisms from Dual Numbers

Definitions and mathematical statement

Let $d\geq3$, let $R$ be a reduced commutative ring containing ${\mathbb{Q}}$, and put $Y=(y_1,\ldots,y_{d-1})$, $A=R[t][x,Y]$, $A_0=A/(t^2)$. Reduced means that $R$ has no nonzero nilpotent elements. The exact source asks for \[ \bar\gamma\in\operatorname{Aut}_x(A_0) \cap\operatorname{Aut}_{R[t]}(A), \qquad \bar\gamma\notin\operatorname{Aut}_x(A). \] This expression is retained, but it compares automorphisms of different rings without specifying an identification, reduction preimage or image convention. The subscript $x$ is also not defined.

There is a canonical reduction homomorphism \[ \rho:\operatorname{Aut}_{R[t]}(A) \longrightarrow\operatorname{Aut}_{R[t]/(t^2)}(A_0), \] since coefficient-fixing automorphisms preserve $(t^2)$. Under the common convention that $\operatorname{Aut}_x$ means fixing the coordinate $x$, two possible questions are quite different:

1. Does some lift $\gamma$ have $\rho(\gamma)\in\operatorname{Aut}_x(A_0)$ but $\gamma\notin\operatorname{Aut}_x(A)$?
2. Does some $\bar\gamma\in\operatorname{Aut}_x(A_0)\cap\operatorname{im}\rho$ admit no lift fixing $x$, that is $\bar\gamma\notin\rho(\operatorname{Aut}_x(A))$?

The first, if read literally with no further condition, is trivial: $\gamma(x)=x+t^2$, $\gamma(Y)=Y$ is an $R[t]$-automorphism reducing to the identity but not fixing $x$. It does not witness the second question, because the identity also has an $x$-fixing lift. Thus an individual non-fixing lift and an obstruction to every fixing lift must not be confused. Another meaning of $\operatorname{Aut}_x$ could change both formulations.

No choice among these interpretations is promoted as the intended conjecture, and no missing tame, congruence, localization or other group condition is guessed. The supplied literature triage checked 21 August 2026 reports no authoritative complete resolution of the extracted claim but does not repair the mixed-ring notation or define the subscript. Source review is necessary before this can specify a nontrivial lifting problem.

Short English statement

The source mixes automorphism groups over $R[t]$ and over its dual-number quotient without defining their comparison or the subscript $x$. A non-fixing lift is different from the absence of every fixing lift. Under a literal fixed-coordinate interpretation the first is trivial, so the intended lifting obstruction cannot be silently inferred.

Sources

• Local UnsolvedMath registry, permanent ID 30000643, exact source code catalogue entry; Lifting Polynomial Automorphisms from Dual Numbers. Proposed year 2007, status open. Corrected_verified editing only balances the given groups/intersection/nonmembership; it supplies no reduction-map or subscript definition.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/01, pp. 5--82, https://doi.org/10.4171/owr/2007/01.
• Supplied literature triage checked 2026-08-21 cites the same report, https://doi.org/10.4171/owr/2007/01, and reports no authoritative complete resolution found. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; no precise lifting theorem or notation repair is supplied.