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Maubach–Willems finite-generation conjecture

For every finite field kk of characteristic p>0p>0 and every integer n≥3n\ge 3, the tame polynomial automorphism group TA⁡n(k)\operatorname{TA}_n(k) is not finitely generated; equivalently, it cannot be generated by the affine subgroup together with finitely many non-affine tame automorphisms.

References

Primary source

arXiv

Additional references

algebragroup-theorypolynomial-automorphisms
Posted

Progress summary

Refreshed
Claimed solved

A new unrefereed preprint claims to settle the conjecture by proving the relevant tame groups are not finitely generated, but the result has not been independently checked.

Maubach and Willems conjectured that, in positive characteristic and dimensions three and higher, the tame polynomial automorphism group cannot be generated by the affine group together with finitely many non-affine automorphisms. Their work supplied an infinite generating set and left finite generation as an open question.

Known results

  • Maubach and Willems, 2011: constructed an infinite set of non-affine generators in dimensions three and higher and conjectured that no finite replacement exists.
  • The characteristic-zero finite-generation theorem of Derksen does not extend in general to characteristic pp.

August 25, 2026 claimed proof

The preprint Iterated Cartier Flux and Non-Finite Generation of Tame Polynomial Automorphism Groups over Finite Fields claims infinite-dimensional abelianized mod-pp quotients in the relevant dimensions and deduces the Maubach–Willems finite-generation classification. This is a claimed resolution, but the manuscript is unrefereed and no independent verification was found.

Current status (as of August 2026): The conjecture has a new claimed proof via infinite-dimensional abelianized mod-pp quotients, but it remains unverified; absent confirmation, the finite-generation classification is not settled.

Sources

Solutions 0

No solutions have been posted yet.