Surjectivity of Reduction Maps for Special Polynomial Automorphisms

Definitions and mathematical statement

Let $R$ be a commutative unital ring containing ${\mathbb{Q}}$, let $n,m\in{\mathbb{Z}}_{>0}$, and let $t,y_1,\ldots,y_n$ be commuting indeterminates. Put $A=R[t]$, $\overline A=R[t]/(t^m)$, and $Y=(y_1,\ldots,y_n)$. For a commutative ring $B$, define \[ \operatorname{SAut}_{B}(B[Y])= \left\{F\in\operatorname{Aut}_{B}(B[Y]): \det\left(\frac{\partial F_i}{\partial y_j}\right)_{i,j=1}^n=1\right\}, \] where $F_i=F(y_i)$ and the determinant equality is in $B[Y]$. Thus $F$ must already be an algebra automorphism, not just a polynomial endomorphism with unit Jacobian. Its inverse is also polynomial. Differentiation is only in the $Y$-variables; all base elements, including $t$, are fixed.

Coefficient reduction modulo $t^m$ induces \[ \rho_{R,n,m}:\operatorname{SAut}_{R[t]}(R[t][Y]) \longrightarrow \operatorname{SAut}_{\overline A}(\overline A[Y]). \] Reducing an automorphism and its inverse preserves the inverse identities, and the Jacobian determinant reduces to one, so this is the stated map.

The question is whether $\rho_{R,n,m}$ is surjective for every stated $R,n,m$. Explicitly, for each $\overline F$ in the target, must there exist $F$ in the source such that \[ F_i\bmod t^m=\overline F_i\quad\text{for }1\leq i\leq n? \] The lift must have determinant exactly one in $R[t][Y]$, rather than just congruent to one, and must be a genuine polynomial automorphism over $R[t]$. A formal $t$-adic lift, a polynomial endomorphism without an inverse, or a lift after adding variables would not answer the question. There is no bound on the degrees of a lift in the supplied wording.

No domain, field, reducedness, Noetherianity or finite-generation hypothesis on $R$ is inserted; the source explicitly permits a general commutative ring containing ${\mathbb{Q}}$. The number of variables remains $n$. The literature triage checked 2026-08-21 reports no authoritative complete resolution located. The open label and notation-only verification are dated source metadata, not an independently checked current theorem.

Short English statement

Must every special polynomial automorphism over the truncated coefficient ring lift to a special polynomial automorphism over $R[t]$, for every commutative ${\mathbb{Q}}$-containing ring? The lift must be polynomially invertible, preserve the same variables and have Jacobian exactly one. Formal or stabilized lifting is a different assertion.

Sources

• Local UnsolvedMath registry, permanent ID 30000644, source code catalogue entry, "Surjectivity of Reduction Maps for Special Polynomial Automorphisms," including original and cleaned map, special-automorphism definition and verification metadata.
• 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 also the sole supplied literature reference.
• Literature triage checked 2026-08-21; local record updated 2026-09-16. No external source has been visited or independently checked here.