Vénéreau polynomials and related fiber bundles

Shulim Kaliman, Mikhail Zaidenberg

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

The Vénéreau polynomials vn:y+xn(xz+y(yu+z2)), n≥1, on A 4 have all fibers isomorphic to the affine space A3. Moreover, for all n≥1 the map (vn,x):A4→ A2 yields a flat family of affine planes over A2. In the present note we show that over the punctured plane A2\{0̄}, this family is a fiber bundle. This bundle is trivial if and only if vn is a variable of the ring ℂ[x][y,z,u] over ℂ[x]. It is an open question whether v1 and v2 are variables of the polynomial ring ℂ[4]=ℂ[x,y,z,u], whereas Vénéreau established that vn is indeed a variable of ℂ[x][y,z,u] over ℂ[x] for n≥3. In this note we give another proof of Vénéreau's result based on the above equivalence. We also discuss some other equivalent properties, as well as the relations to the Abhyankar-Sathaye Embedding Problem and to the Dolgachev-Weisfeiler Conjecture on triviality of flat families with fibers affine spaces.

Original languageEnglish (US)
Pages (from-to)275-286
Number of pages12
JournalJournal of Pure and Applied Algebra
Volume192
Issue number1-3
DOIs
StatePublished - Sep 1 2004

ASJC Scopus subject areas

  • Algebra and Number Theory

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