TY - JOUR
T1 - Vénéreau polynomials and related fiber bundles
AU - Kaliman, Shulim
AU - Zaidenberg, Mikhail
N1 - Copyright:
Copyright 2004 Elsevier B.V., All rights reserved.
PY - 2004/9/1
Y1 - 2004/9/1
N2 - The Vénéreau polynomials vn:y+xn(xz+y(yu+z2)), n≥1, on Aℂ 4 have all fibers isomorphic to the affine space Aℂ3. Moreover, for all n≥1 the map (vn,x):Aℂ4→ Aℂ2 yields a flat family of affine planes over Aℂ2. In the present note we show that over the punctured plane Aℂ2\{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.
AB - The Vénéreau polynomials vn:y+xn(xz+y(yu+z2)), n≥1, on Aℂ 4 have all fibers isomorphic to the affine space Aℂ3. Moreover, for all n≥1 the map (vn,x):Aℂ4→ Aℂ2 yields a flat family of affine planes over Aℂ2. In the present note we show that over the punctured plane Aℂ2\{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.
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U2 - 10.1016/j.jpaa.2004.01.009
DO - 10.1016/j.jpaa.2004.01.009
M3 - Article
AN - SCOPUS:2942683252
VL - 192
SP - 275
EP - 286
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
SN - 0022-4049
IS - 1-3
ER -