VÉNÉREAU POLYNOMIALS AND RELATED FIBER BUNDLES

SHULIM KALIMAN AND MIKHAIL ZAIDENBERG

Abstract. The Vénéreau polynomials

v_{n} := y + x^{n}(xz + y(yu + z^{2})), n ≥ 1,

on 𝔸_{ℂ}^{4} have all fibers isomorphic to the affine space 𝔸_{ℂ}^{3}. Moreover, for all n ≥ 1 the map (v_{n}, x) : 𝔸_{ℂ}^{4} → 𝔸_{ℂ}^{2} yields a flat family of affine planes over 𝔸_{ℂ}^{2}. In the present note we show that over the punctured plane 𝔸_{ℂ}^{2}\{¯0}, this family is a fiber bundle. This bundle is trivial if and only if v_{n} is a variable of the ring ℂ[x][y, z, u] over ℂ[x].

It is an open question whether v_{1} and v_{2} are variables of the polynomial ring ℂ^{[4]} = ℂ[x, y, z, u], whereas S. Vénéreau established that v_{n} 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.

In [KVZ1, KVZ2] polynomials in four variables of the form

p = f(x, y)u + g(x, y, z) (1)

were studied. It was shown that p : 𝔸_{ℂ}^{4} → 𝔸_{ℂ}^{1} is a flat family of affine spaces (i.e., every fiber p^{∗}(c), c ∈ 𝔸_{ℂ}^{1}, is reduced and isomorphic to 𝔸_{ℂ}^{3}) provided that p^{∗}(0) ≃ 𝔸_{ℂ}^{3} ([KVZ2, Theorem 3.21]). As for the latter condition, a criterion is given in terms of the morphism π : C → Γ, where C = {f = g = 0} ⊆ 𝔸_{ℂ}^{3} and Γ = {f = 0} ⊆ 𝔸_{ℂ}^{2} are affine curves and π is the projection (x, y, z) ⟼ (x, y) [KVZ2, Theorems 2.11 and 3.21]. Based on this, S. Vénéreau [Ve] considered the following polynomials generating flat families of affine spaces:

v_{n} := y + x^{n}(xz + y(yu + z^{2})) = x^{n}y^{2}u + y + x^{n+1}z + x^{n}yz^{2}, n ≥ 1 . (2)

He showed that for every n ≥ 3, v_{n} is a variable of the ring ℂ^{[4]}, or in other words, a coordinate of a polynomial automorphism of 𝔸_{ℂ}^{4}. Furthermore, some sufficient conditions were found (see [KVZ2, Section 4.1]) for a polynomial p of form (1) to be a variable. However we do not know whether the latter is true if these sufficient conditions are not satisfied, and this is the case for v_{1} and v_{2}. That is, we do not know whether the hypersurfaces v_{i}^{−1}(0) ≃ 𝔸_{ℂ}^{3}, i = 1, 2, in 𝔸_{ℂ}^{4} can be rectified by means of polynomial automorphisms of 𝔸_{ℂ}^{4} (this is a particular case of the Abhyankar-Sathaye Embedding Problem).

It was also shown in [Ve] that for every n ≥ 3 the morphism

Φ_{n} = (v_{n}, x) : 𝔸_{ℂ}^{4} → 𝔸_{ℂ}^{2}

Acknowledgments: This research started during a visit of the first author at the Institut Fourier of the University of Grenoble, and continued during a stay of the second author at the Max Planck Institute of Mathematics at Bonn. The authors thank both institutions for their support. It is our pleasure to thank Don Zagier for stimulating discussions, and M. Uludag for his help with MAPLE.

1991 Mathematics Subject Classification: 14R10, 14R25. Key words: polynomial ring, variable, algebraic fiber bundle, flat family.

defines a trivial family of affine planes. This means that the Vénéreau polynomials v_{n} (n ≥ 3) are x-variables i.e., variables of the ℂ[x]-algebra ℂ[x][y, z, u].

If a polynomial p as in (1) defines a flat family of affine spaces p : 𝔸_{ℂ}^{4} → 𝔸_{ℂ}^{1} then p also defines, along with a suitable second polynomial q ∈ ℂ[x, y] ⊆ ℂ^{[4]}, which is a variable in ℂ[x, y], a flat family of affine planes Φ = (p, q) : 𝔸_{ℂ}^{4} → 𝔸_{ℂ}^{2} (see [KVZ2, Theorem 3.21]). Thus the question arises whether the two remaining families of affine planes Φ_{n} = (v_{n}, x) : 𝔸_{ℂ}^{4} → 𝔸_{ℂ}^{2} (n = 1, 2) and, more generally, all such families of affine planes Φ = (p, q) : 𝔸_{ℂ}^{4} → 𝔸_{ℂ}^{2} as above, are trivial.

In the present note we address the former special question by showing in Proposition 1 that Φ_{n} restricts to an algebraic fiber bundle, say, λ_{n} over the punctured plane:

λ_{n} := Φ_{n}|(𝔸_{ℂ}^{4}\𝔸_{ℂ}^{2}) : 𝔸_{ℂ}^{4}\𝔸_{ℂ}^{2} → 𝔸_{ℂ}^{2}\{¯0} ,

where 𝔸_{ℂ}^{2} ⊆ 𝔸_{ℂ}^{4} is the coordinate (z, u)-plane. This bundle has 𝔸_{ℂ}^{2} as the typical fiber and Aut(𝔸_{ℂ}^{2}) as the structure group. We show that λ_{n} is trivial for every n ≥ 3, thus recovering Vénéreau’s result (see Corollary 1 below). In the cases n = 1, 2 we were not able to carry the computations needed by our methods, and so the question remains open.

Generally speaking, we deal below with the following three categories of affine Sschemes f : X → S over a quasiprojective base S:

• The flat families of affine m-spaces over S ^{1};

• The algebraic fiber bundles over S with fiber 𝔸_{ℂ}^{m};

• The algebraic vector bundles of rank m over S. The Dolgachev-Weisfeiler Conjecture [DW, (3.8.3)] claims that the first category reduces to the second one. In turn, for an affine base S, the second one reduces to the third one due to the Bass-Connell-Wright Theorem [BCW].

If the Dolgachev-Weisfeiler Conjecture were answered in affirmative this would provide an affirmative answer to our general question. Indeed, in this case Φ (in particular, Φ_{n}) would be a fiber bundle over 𝔸_{ℂ}^{2} with fiber 𝔸_{ℂ}^{2}, thus a vector bundle by the Bass-Connell-Wright Theorem, hence it must be trivial due to the Quillen-Suslin Theorem.

Conversely, if any one of the bundles λ_{1} or λ_{2} were non-trivial this would provide a counterexample to the Dolgachev-Weisfeiler Conjecture (with m = 2), and, presumably, to the Abhyankar-Sathaye Embedding Problem.

Summarizing, the following equivalences hold. Proposition 1. If v ∈ ℂ^{[4]} is a polynomial such that Φ = (x, v) yields a flat family of affine planes Φ : 𝔸_{ℂ}^{4} → 𝔸_{ℂ}^{2} then the following conditions are equivalent:

(i) v is an x-variable of the ring ℂ[x, y, z, u].

(ii) Φ : 𝔸_{ℂ}^{4} → 𝔸_{ℂ}^{2} is an algebraic fiber bundle.

(iii) It is a trivial fiber bundle.

(iv) λ := Φ|(𝔸_{ℂ}^{4}\F¯0), where F¯0 = Φ^{−1}(¯0), is a trivial fiber bundle over 𝔸_{ℂ}^{2}\{¯0}. Proof. The equivalence (i)⇔(iii) is a tautology, and (iii)⇔(ii) follows by combining the Bass-Connell-Wright Reduction Theorem and the Quillen-Suslin Theorem as above. The implication (iii)⇒(iv) is evident, whereas the converse one (iv)⇒(iii) can be easily obtained by extending the trivialization morphism

𝔸_{ℂ}^{4}\F¯0 → (𝔸_{ℂ}^{2}\{¯0}) × 𝔸_{ℂ}^{2}

1That is, every fiber X_{s} = f^{∗}(s), s ∈ S, is a reduced scheme isomorphic to 𝔸_{ℂ}^{m}.

VÉNÉREAU POLYNOMIALS AND RELATED FIBER BUNDLES 3

and its inverse to the deleted planes F¯0 and {¯0} × 𝔸_{ℂ}^{2}, respectively. □

Notation 1. We use below the following polynomials from the ring ℂ[x, y, z, u] :

(3)  w  = z^{2} + yu ,
(4)  t  = xz + yw ,
(5)  s  = −(2xz + yw)w ,
(6)  η  = s + x^{2}u = x^{2}u − 2xzw − yw^{2} ,
(7)  ζ^{(1)} = −v_{1}z + v_{1}t(v_{1}u + z^{2} + w) + t^{2}(xz^{2} + st) ,
(8)  ζ^{(2)}  = −z + xt(v_{2}u + z^{2} + st + w) ,
(9)  ζ^{(n)} = −z + x^{n−3}t (v_{n}η + x^{2}w) ,  n ≥ 3 ,

where v_{n} = y + x^{n}t (10) stands for the Vénéreau polynomials (2). We also consider the rational functions ξ_{0} = η/x^{3}, ξ_{1}^{(1)} = ζ^{(1)}/v_{1}^{3} and ξ_{1}^{(n)} = ζ^{(n)}/v_{n}^{2}, n ≥ 2 . (11) We let

L_{0} = ℂ[x, x^{−1}], L_{1} = ℂ[v, v^{−1}], K_{0} = C[x, x^{−1}, v], K_{1} = ℂ[x, v, v^{−1}] ,
R = ℂ[x, v] = K_{0} ∩ K_{1} and M = ℂ[x, x^{−1}, v, v^{−1}] ,

with the convention that v = v_{n} whenever we consider Φ_{n} or λ_{n}. Letting 𝔸_{ℂ}^{2} = Spec ℂ[x, v_{n}], the punctured plane 𝔸_{ℂ}^{2}\{¯0} can be covered by the Zariski open subsets

U_{0} = Spec K_{0} = 𝔸_{ℂ}^{2}\{x = 0} and U_{1} = Spec K_{1} = 𝔸_{ℂ}^{2}\{v_{n} = 0} , where U_{0} ∩ U_{1} = Spec M.

With this notation the following results hold. Proposition 2. (a) The morphism φ_{0} = (t, ξ_{0}), respectively, φ_{1}^{(n)} = (t, ξ_{1}^{(n)}), yields a local trivialization for the family λ_{n} over U_{0}, respectively, over U_{1}. Thus λ_{n} is an algebraic fiber bundle with 𝔸_{ℂ}^{2} as the typical fiber.

(b) The transition function

φ_{10}^{(n)} = φ_{1}^{(n)} ∘ φ_{0}^{−1} : (t, ξ_{0}) ⟼ (t, ξ_{1}^{(n)})

is a triangular automorphism with

ξ_{1}^{(n)} = ξ_{0} + \frac{p_{n}(t)}{x^{k}v_{n}^{l}},  where  (k, l) = \begin{cases}(3, 2), & n ≥ 2 \\ (3, 3), & n = 1 , \end{cases} (12)

and the polynomials p_{n} ∈ ℂ[x, v_{n}, t] are given by

p_{n}(t) = \begin{cases}v_{n}t^{2} − x^{2}t, & n ≥ 3 \\ x^{2}t^{3} + v_{2}t^{2} − x^{2}t, & n = 2 \\ x^{2}t^{4} + xv_{1}t^{3} + v_{1}^{2}t^{2} − x^{2}v_{1}t, & n = 1 . \end{cases} (13)

Proof. (a) The Nagata automorphism α ∈ Aut(𝔸_{ℂ}^{3}) of 𝔸_{ℂ}^{3} = Spec ℂ[y, z, u] is given (see [Na]) by α : (y, z, u) → (y, t_{0}, η_{0}), where

t_{0} = z + yw and η_{0} = u − 2zw − yw^{2}

with w = z^{2} +yu as in (3) above. ^{2} Composing α with the following L_{0}-automorphisms of L_{0}[y, z, u] (cf. [Ve]):

g : (y, z, u) ⟼ (x^{−2}y, xz, x^{4}u) and h : (y, z, u) ⟼ (x^{2}y, z, x^{−2}u)

we obtain:

w ∘ g = x^{2}w, t_{0} ∘ g = t, η_{0} ∘ g = x^{2}η

and

β := h ∘ α ∘ g ∈ Aut_{L_{0}}(L_{0}[y, z, u]), where β : (y, z, u) ⟼ (y, t, η) .

Letting

γ_{n} : (y, z, u) ⟼ (v_{n} = y + x^{n}t, t, η)

it follows that γ_{n} ∘ β ∈ Aut_{L_{0}}(L_{0}[y, z, u]). Thus

Φ_{n}^{−1}(U_{0}) = 𝔸_{ℂ}^{4}\{x = 0} = Spec ℂ[x, x^{−1}, y, z, u] = Spec ℂ[x, x^{−1}, v_{n}, t, η] ≅ U_{0} × 𝔸_{ℂ}^{2}

with 𝔸_{ℂ}^{4} = Spec ℂ[x, y, z, u] and 𝔸_{ℂ}^{2} = Spec ℂ[t, η]. Hence φ_{0} = (t, ξ_{0}) = (t, η/x^{3}) yields indeed a trivialization of λ_{n} over U_{0}.

To show that also φ_{1}^{(n)} = (t, ξ_{1}^{(n)}) yields a trivialization of λ_{n} over U_{1}, we consider separately the cases n = 1, n = 2 and n ≥ 3. We will constantly exploit the relations

ys + t^{2} = x^{2}z^{2}  ⇒  v_{n}s + t^{2} = x^{2}z^{2} + x^{n}st (14)

⇒ v_{n}η + t^{2} = x^{2}(z^{2} + v_{n}u) + x^{n}st (15)

(see (6) and (10)).

Case n = 1. We denote

ζ′ := \frac{v_{1}η + t^{2}}{x} = x(v_{1}u + z^{2}) + st (16)

(see (15)),

ζ′′ := \frac{v_{1}ζ′ + t^{3}}{x} = v_{1}(v_{1}u + z^{2}) + xz^{2}t + st^{2} (17)

= yw + xtw + xtv_{1}u + xz^{2}t + st^{2}

(see (3) and (10)), and

ζ′′′ := ζ′′ − t = −xz + xtw + xtv_{1}u + xz^{2}t + st^{2} (18)

δ_{1} : (y, z_{1}, u_{1}) ⟼ (y, z_{1}, w_{1}) , w_{1} := u_{1} + z_{1}^{2},

δ_{2} : (y, z_{1}, w_{1}) ⟼ (y, t_{1}, w_{1}) , t_{1} := z_{1} + y^{2}w_{1},
δ_{3} : (y, t_{1}, w_{1}) ⟼ (y, t_{1}, η_{1}) , η_{1} := w_{1} − t_{1}^{2} .

2Let us observe in passing that α = μ ∘ δ ∘ μ^{−1}, where μ : (y, z_{1}, u_{1}) ⟼ (y, z = yz_{1}, u = yu_{1}), or else μ : (y, t_{1}, η_{1}) ⟼ (y, t_{0} = yt_{1}, η_{0} = yη_{1}) is a birational morphism, or in other words, an affine modification of 𝔸_{ℂ}^{3} with the locus (D, ¯0), where D = {y = 0}, and δ := δ_{3} ∘ δ_{2} ∘ δ_{1} ∈ Aut(𝔸_{ℂ}^{3}) is a tame ℂ[y]-automorphism with

(see (4)). Now

ζ^{(1)} := \frac{v_{1}ζ′′′ + t^{4}}{x} = \frac{v_{1}^{3}η + p_{1}(t)}{x^{3}} (19)

verifies both (7) and (12) for n = 1.

Further, for any point C = (c_{1}, c_{2}) ∈ 𝔸_{ℂ}^{2} with c_{1} ≠ 0, c_{2} ≠ 0, the functions (t, η), hence also (t, ζ^{(1)}), give global coordinates on the fiber F_{C} = Φ_{1}^{−1}(C) ≃ 𝔸_{ℂ}^{2} over C. In the case c_{1} = 0, c_{2} ≠ 0 the following hold:

v_{1} = y = c_{2}, w = c_{2}^{−1}t = c_{2}u + z^{2}, η = s = −c_{2}^{−1}t^{2} , (20)

and

ζ^{(1)} = −c_{2}z + 2t^{2} − c_{2}^{−1}t^{5} (21)

(see (3), (7), (10) and (14)). Therefore we obtain:

ℂ[z, u] = ℂ[z, t] = ℂ[t, ζ^{(1)}] = ℂ[t, ξ_{1}^{(1)}] . (22)

Clearly, F_{C} ≃ 𝔸_{ℂ}^{2} = Spec ℂ[z, u] for C = (0, c_{2}) with c_{2} ≠ 0. In other words, (z, u) give coordinates on the fiber F_{C}. Thus by (22) for c_{2} ≠ 0 the functions (t, ζ^{(1)}), and hence also (t, ξ_{1}^{(1)}), provide as well coordinates on F_{C}. Now (a) follows for n = 1.

Case n = 2. From (15) we obtain:

ζ′ := \frac{v_{2}η + t^{2}}{x^{2}} = z^{2} + uv_{2} + st . (23)

It can be easily seen that the polynomial

ζ^{(2)} := \frac{v_{2}ζ′ + t^{3} − t}{x} = \frac{v_{2}^{2}η + p_{2}(t)}{x^{3}} (24)

verifies both (8) and (12) for n = 2. It follows that for C = (c_{1}, c_{2}) with c_{1} ≠ 0, c_{2} ≠ 0, the map (t, η) ⟼ (t, ζ^{(2)}) provides an isomorphism

F_{C} ≃ 𝔸_{ℂ}^{2} = Spec ℂ[t, η] = Spec ℂ[t, ζ^{(2)}] .

As above, for c_{1} = 0, c_{2} ≠ 0 we get

ℂ[z, u] = ℂ[z, t] = ℂ[t, ζ^{(2)}]

and so, the functions (t, ζ^{(2)}) still give coordinates on the fiber F_{C}. This proves (a) for n = 2.

Case n ≥ 3. Letting again

ζ′ := \frac{v_{n}η + t^{2}}{x^{2}} = z^{2} + v_{n}u + x^{n−2}st (25)

(see (15)) we can easily see that

ζ^{(n)} := \frac{v_{n}ζ′ − t}{x} = \frac{v_{n}^{2}η + p_{n}(t)}{x^{3}} (26)

verifies both (9) and (12), the map (t, η) ⟼ (t, ζ^{(n)}) provides an isomorphism F_{C} ≃ 𝔸_{ℂ}^{2} = Spec ℂ[t, ζ^{(n)}] as soon as c_{1} ≠ 0 and c_{2} ≠ 0, and, moreover, the functions (t, ζ^{(n)}) yield coordinates on any fiber F_{C} with c_{2} ≠ 0. This proves (a) in the general case. Actually (b) has been established in the course of proof of (a). □

Remark 1. By Proposition 2(b) the transition function φ_{10}^{(n)} of λ^{(n)} as in (12) takes values in the subgroup of plane triangular automorphisms of the form

(t, ξ) ⟼ (t, ξ + \frac{q(t)}{x^{n}v^{m}} ) with q ∈ R[t], n, m ∈ ℕ ∪ \{0\} .

However, within this subgroup, the fiber bundles λ_{n} are non-trivial. Indeed, triviality of λ_{n} is equivalent to the existence of a decomposition

\frac{p_{n}(t)}{x^{k}v^{l}} = \frac{a_{n}(t)}{x^{k+α}} − \frac{b_{n}(t)}{v^{l+β}} with a_{n}, b_{n} ∈ R[t], α, β ≥ 0 ,

and hence is equivalent to: p_{n} ∈ (x^{k}, v^{l})R[t], which is not the case (see (13)).

The following results will be useful for establishing the triviality of bundles λ_{n}.

Proposition 3. Let λ be an algebraic fiber bundle over 𝔸_{ℂ}^{2}\{¯0}, where 𝔸_{ℂ}^{2} = Spec ℂ[x, v], with 𝔸_{ℂ}^{2} = Spec ℂ[t, ξ] as the typical fiber. We suppose that the restrictions λ|U_{i} (i = 0, 1) are trivial. If the transition function

φ_{10} : U_{0} ∩ U_{1} → Aut(ℂ[t, ξ])

has the form:

φ_{10} : (t, ξ) ⟼ (t, ξ + \frac{p(t)}{x^{k}v^{l}} ) (27)

with p ∈ R[t], where R = ℂ[x, v] and k, l ∈ ℕ, then the following hold.

(a) λ is trivial if and only if

φ_{10} = τ_{1} ∘ τ_{0}^{−1}  with  τ_{i} ∈ Aut_{K_{i}}K_{i}[t, ξ], i = 0, 1 . (28)

If (28) is fulfilled then necessarily τ_{0} and τ_{1} have the form:

τ_{0} : (t, ξ) ⟼ (a, \frac{b_{0}}{x^{k}} )  and  τ_{1} : (t, ξ) ⟼ (a, \frac{b_{1}}{v^{l}} ) , (29)

where a, b_{0}, b_{1} ∈ R[t, ξ] satisfy the cocycle relation

x^{k}b_{1} − v^{l}b_{0} = p(a) . (30)

Furthermore, up to multiplying b_{0}, b_{1} by a constant we may assume that the following Jacobian relations hold:

(31) jac (τ_{0}) = 1 ⇔ jac (a, b_{0}) = x^{k}, jac (τ_{1}) = 1 ⇔ jac (a, b_{1}) = v^{l} ,

where jac (∗, ∗) stands for the jacobian in t, ξ.

(b) Any solution (a, b_{0}, b_{1}) in (R[t, ξ])^{3} of (30) satisfies

\frac{jac (a, b_{0})}{x^{k}} = \frac{jac (a, b_{1})}{v^{l}} =: d ∈ R (32)

and

jac (b_{0}, b_{1}) = −d · p′(a) . (33)

Such a solution also verifies (31) if and only if d = 1. In the latter case

jac (b_{0}, b_{1}) = −p′_{t}(a) . (34)

Proof. Although (a) is well known, we remind the proof. We have: φ_{10} = φ_{1} ∘ φ_{0}^{−1}, where φ_{i} is a trivialization of λ|U_{i} (i = 0, 1). If (28) holds then ψ|U_{i} := τ_{i}^{−1}φ_{i} (i = 0, 1) defines a global trivialization of λ. Indeed, ψ is a well defined regular map over U_{0} ∪U_{1} = 𝔸_{ℂ}^{2}\{¯0}, since by (28), τ_{0}^{−1}φ_{0} = τ_{1}^{−1}φ_{1} in U_{0} ∩U_{1}. Conversely, if ψ is a global trivialization of λ then τ_{i} := φ_{i} ∘ψ^{−1} ∈ Aut_{K_{i}}K_{i}[t, ξ] (i = 0, 1) satisfy (28), as required.

Letting τ_{i} = (â_{i}, b̂_{i}) with â_{i}, b̂_{i} ∈ K_{i}[t, ξ], by (28) we obtain:

(â_{1}, b̂_{1}) = φ_{10} ∘ (â_{0}, b̂_{0}) = (â_{0}, b̂_{0} + \frac{p(â_{0})}{x^{k}v^{l}}) . (35)

Therefore â_{0} = â_{1} =: a ∈ R[t, ξ], and

b̂_{0} = \frac{b_{0}}{x^{k+α}} ,  b̂_{1} = \frac{b_{1}}{v^{l+β}} , (36)

where b_{0}, b_{1} ∈ R[t, ξ] and α, β ≥ 0. Now (36) yields:

x^{k+α}b_{1} − v^{l+β}b_{0} = x^{α}v^{β}p(a) , (37)

hence x^{α}|b_{0} and v^{β}|b_{1}. Thus without loss of generality we can suppose that α = β = 0, and so (30) follows.

As jac (τ_{i}) is a unit in K_{i}[t, ξ] (i = 0, 1) we have

jac (τ_{0}) = c_{0}x^{n} and jac (τ_{1}) = c_{0}v^{m} for some n, m ∈ ℤ and c_{0}, c_{1} ∈ ℂ\{0} .

Since jac (φ_{10}) = 1 it follows from (28) that

jac (τ_{0}) = jac (τ_{1}) ∈ K_{0} ∩ K_{1} = R

is a unit in R. Hence n = m = 0, and we may suppose that c_{0} = c_{1} = 1, which yields (31).

(b) Applying to (30) the derivations ∗ ⟼ jac (a, ∗) and ∗ ⟼ jac (∗, b_{1}), respectively, gives

(38) x^{k}jac (a, b_{1}) = v^{l}jac (a, b_{0}) and p′(a)jac (a, b_{1}) = −v^{l}jac (b_{0}, b_{1}) .

Now (32) and (33) follow. □

Remarks 2. 1. If (a, b_{0}, b_{1}) is a solution of (30) then so is (a, b_{0} + x^{k}c, b_{1} + v^{l}c) for any c ∈ R[t, ξ]. This new solution verifies (31) if and only if jac (a, c) = 1 − d with d as in (32).

Similarly, (a + A, b_{0} + B_{0}, b_{1} + B_{1}) with A, B_{0}, B_{1} ∈ R[t, ξ] gives a new solution of (30) if and only if

∑_{j=1}^{deg p} \frac{p^{(j)}(a)}{j!}A^{j} = x^{k}B_{1} − v^{l}B_{0} . (39)

2. By virtue of (32), the cocycle relation (30) and one of the Jacobian relations (31) imply the other one.

3. Modulo the plane Jacobian Conjecture the maps τ_{0} and τ_{1} as in (29) are invertible if and only if the Jacobian relations (31) hold. Presumably, in our particular setting the degrees of the coordinate polynomials can be limited to the range where the plane Jacobian Conjecture is known to be true. If so then the triviality of the bundles λ_{n} reduces to the existence of a solution (a, b_{0}, b_{1}) ∈ (R[t, ξ])^{3} of both (30) and (31).

8 SHULIM KALIMAN AND MIKHAIL ZAIDENBERG

Lemma 1. Suppose that p = p_{n} in (30) and (k, l) = (3, 2) if n ≥ 2, (k, l) = (3, 3) if n = 1. Then there exists a solution (a, b_{0}, b_{1}) ∈ (R[t, ξ])^{3} of (30) with

a = ∑_{i,j≥0} a_{ij}x^{i}v^{j}, where a_{ij} ∈ S := ℂ[t, ξ] , (40)

if and only if a_{00} = 0 and a_{01} = a_{10}^{2} . (41) Proof. Clearly, (30) has a solution if and only if

p(a) = p_{n}(a) ∈ (x^{k}, v^{l}), where (k, l) = \begin{cases}(3, 2), & n ≥ 2 \\ (3, 3), & n = 1 . \end{cases} (42)

We have

p_{n}(a) ≡ \begin{cases}a_{00}^{2}v mod (x, v^{2}), & n ≥ 2 \\ a_{00}^{3}xv mod (x^{2}, v^{2}), & n = 1 . \end{cases} (43)

In any case p_{n}(a) ∈ (x^{k}, v^{l}) ⇒ a_{00} = 0 , (44) and then

p_{n}(a) ≡ \begin{cases}(a_{10}^{2} − a_{01})x^{2}v mod (x^{3}, v^{2}), & n ≥ 2 \\ (a_{10}^{2} − a_{01})x^{2}v^{2} mod (x^{3}, v^{3}), & n = 1 . \end{cases} (45)

Now the assertion follows. □ Notation 2. Letting in Proposition 3

p(t) = ∑_{j=0}^{deg p} r_{j}t^{j}, where r_{j} ∈ R = ℂ[x, v] ,

and supposing that r_{0} = 0, we can define the successive approximations

φ_{10}^{(m)} : (t, ξ) ⟼ (t, ξ + \frac{∑_{j=0}^{m} r_{j}t^{j}}{x^{k}v^{l}} ) , m = 1, …, deg p, (46)

to our transition function φ_{10}. These φ_{10}^{(m)} may be considered as the transition functions of a succession of algebraic fiber bundles λ^{(m)} over the punctured plane 𝔸_{ℂ}^{2}\{¯0} with 𝔸_{ℂ}^{2} as the typical fiber. In particular, the linear part φ_{10}^{(1)} defines a rank 2 vector bundle λ^{(1)} over the punctured plane.

With these notation the following holds. Lemma 2. If r_{1} = −x^{k′}v^{l′}, where α := k − k′ ≥ 0 and β := l − l′ ≥ 0, then the vector bundle λ^{(1)} is trivial. In particular, λ_{n}^{(1)} is trivial for every n ≥ 1. Proof. Similarly as in Proposition 3(a), λ^{(1)} is trivial if and only if φ_{10}^{(1)} = τ_{1}^{(1)} ∘ (τ_{0}^{(1)})^{−1} with τ_{i}^{(1)} ∈ GL(2, K_{i}), i = 0, 1 . (47)

VÉNÉREAU POLYNOMIALS AND RELATED FIBER BUNDLES 9

By our assumption the linear part φ_{10}^{(1)} of φ_{10} is:

φ_{10}^{(1)} : (t, ξ) ⟼ (t, ξ − x^{−α}v^{−β}t) .

It can be represented by the following matrix:

g = (\begin{matrix}1 & 0 \\ −x^{−α}v^{−β} & 1 \end{matrix}) ∈ SL(2, M) . (48)

Letting e.g.,

τ_{0}^{(1)} = (\begin{matrix}v^{β} & −x^{α} \\ x^{−α} & 0 \end{matrix}) ∈ SL(2, K_{0}) and τ_{1}^{(1)} = (\begin{matrix}v^{β} & −x^{α} \\ 0 & v^{−β} \end{matrix}) ∈ SL(2, K_{1}) (49)

we obtain g ∘τ_{0}^{(1)} = τ_{1}^{(1)}, which shows that λ^{(1)} is indeed trivial (cf. (28)). According to (13) in Proposition 2, λ = λ_{n} corresponds to the particular case where (α, β) = (1, 2), hence λ_{n}^{(1)} is also trivial. □

The similar triviality result holds also for the second order approximations. Proposition 4. The fiber bundle λ_{n}^{(2)} is trivial for every n ≥ 1. Proof. By virtue of (13) all the fiber bundles λ_{n}^{(2)}, n ≥ 1, actually have the same transition function

φ_{10}^{(2)} : (t, ξ) ⟼ (t, ξ − \frac{t}{xv^{2}} + \frac{t^{2}}{x^{3}v} ) . (50)

If a decomposition φ_{10}^{(2)} = τ_{1}^{(2)} ∘ (τ_{0}^{(2)})^{−1} with τ_{i}^{(2)} ∈ Aut_{K_{i}}K_{i}[t, ξ], i = 0, 1 , (51) as in (29) does exist then clearly there should also exist such a decomposition with the linear parts τ_{i}^{(1)} of τ_{i}^{(2)} (i = 0, 1) as in (49), where (α, β) = (1, 2), i.e., with

a = v^{2}t − xξ + O(2), b_{0} = x^{2}t + O(2) and b_{1} = \begin{cases}ξ + O(2), & n ≥ 2 \\ vξ + O(2), & n = 1 , \end{cases} (52)

where O(2) stands for the terms of order ≥ 2 in t, ξ. By Lemma 1, a_{00} = 0 and a_{01} = a_{10}^{2}. This indicates that the power series expansion of the polynomial a ∈ ℂ[x, v, t, ξ] can start e.g., with: ã = v^{2}t − xξ + vξ^{2} = vδ − xξ , (53) where δ := vt + ξ^{2}, a_{10} = −ξ and a_{01} = a_{10}^{2} = ξ^{2} . (54) For the fiber bundles λ_{n}^{(2)} the cocycle relation (30) becomes (see (13)):

(55)  −x^{2}a + va^{2} = x^{3}b_{1} − v^{2}b_{0},  n ≥ 2
(56)  −x^{2}va + v^{2}a^{2} = x^{3}b_{1} − v^{3}b_{0},  n = 1 .

It is satisfied e.g., by the triple (a, b_{0}, b_{1}) with

a = ã,  b_{0} = x^{2}t − vδ^{2} + 2xδξ  and b_{1} = \begin{cases}ξ, & n ≥ 2 \\ vξ, & n = 1 , \end{cases} (57)

cf. (52) (recall that (k, l) = (3, 2) if n ≥ 2, (k, l) = (3, 3) if n = 1). Thus for any n ≥ 1 one can take e.g.,

τ_{0}^{(2)} : (t, ξ) ⟼ (a, \frac{b_{0}}{x^{k}} ) = (vδ − xξ, \frac{1}{x}t − \frac{v}{x^{3}} δ^{2} + \frac{2}{x^{2}} δξ) (58)

and

τ_{1}^{(2)} : (t, ξ) ⟼ (a, \frac{b_{1}}{v^{l}} ) = (vδ − xξ, \frac{1}{v^{2}} ξ) (59)

with δ = vt + ξ^{2}. It is easily seen that jac (τ_{i}^{(2)}) = 1, i = 0, 1. We have checked with

MAPLE that indeed τ_{i}^{(2)} ∈ Aut_{K_{i}}K_{i}[t, ξ], i = 0, 1, with the inverse map (τ_{1}^{(2)})^{−1} given by:

(τ_{1}^{(2)})^{−1} : (t, ξ) ⟼ (\frac{1}{v^{2}} t + xξ − v^{3}ξ^{2}, v^{2}ξ) . (60)

In view of (50) and (60), from (τ_{0}^{(2)})^{−1} = (τ_{1}^{(2)})^{−1} ∘ φ_{10}^{(2)} we obtain:

(τ_{0}^{(2)})^{−1} : (t, ξ) ⟼ (61)

(xξ − v^{3}ξ^{2} + \frac{2v}{x} tξ − \frac{2v^{2}}{x^{3}} t^{2}ξ + \frac{2}{x^{4}} t^{3} − \frac{v}{x^{6}} t^{4}, v^{2}ξ − \frac{1}{x}t + \frac{v}{x^{3}} t^{2}) .

Thus by Proposition 3(a), λ_{n}^{(2)} is trivial for any n ≥ 1, as stated. □

As a corollary of Propositions 1 and 4 we recover the result of Vénéreau [Ve]:

Corollary 1. For every n ≥ 3, the polynomial v_{n} is an x-variable of the polynomial ring ℂ[x, y, z, u]. More precisely, the map

α_{n} : 𝔸_{ℂ}^{4} → 𝔸_{ℂ}^{4}, (x, y, z, u) ⟼ (x, v_{n}, ζ^{(n)}, θ^{(n)}) ,

is an x-automorphism of the ring ℂ[x, y, z, u], where

θ^{(n)} := u − x^{n−3}t (xw^{2} + η(ζ^{(n)} − z + x^{n−1}tw)) . (62)

Proof. Indeed, in view of (13), for n ≥ 3 we have φ_{10} = φ_{10}^{(2)} and so, λ_{n} = λ_{n}^{(2)} is trivial. Furthermore, by virtue of (60) and Proposition 2(a) the trivialization ψ_{n} of λ_{n}:

ψ_{n} := (τ_{1}^{(2)})^{−1} ∘ φ_{1}^{(n)} = (\frac{t + xζ^{(n)} − v_{n}(ζ^{(n)})^{2}}{v_{n}^{2}} , ζ^{(n)}) =: (θ^{(n)}, ζ^{(n)}) (63)

extends to a morphism

Ψ_{n} : 𝔸_{ℂ}^{4} = Spec ℂ[x, y, z, u] → 𝔸_{ℂ}^{2} = Spec ℂ[θ^{(n)}, ζ^{(n)}]

providing an automorphism α_{n} := (Φ_{n}, Ψ_{n}) ∈ Aut(𝔸_{ℂ}^{4}) (see the proof of Proposition 3). The explicit expression (62) for θ^{(n)} was found by applying formulas (3)-(6), (9)-(10),

(26) and (63); it was checked with MAPLE. □

References

[BCW] H. Bass, E. H. Connell, D. L. Wright, Locally polynomial algebras are symmetric algebras, Invent. Math. 38 (1976/77), 279–299.

[DW] I.V. Dolgachev, B. Ju. Veisfeiler, Unipotent group schemes over integral rings, Math. USSR Izv. 38 (1975), 761–800.

[KVZ1] S. Kaliman, S. Vénéreau, M. Zaidenberg, Extensions birationnelles simples de l’anneau de polynômes ℂ^{[3]}, C. R. Acad. Sci. Paris Sr. I Math. 333 (2001), 319–322.

[KVZ2] S. Kaliman, S. Vénéreau, M. Zaidenberg, Simple birational extensions of the polynomial ring ℂ^{[3]}, E-print math.AG/0104204, 49p. (to appear in Trans. Amer. Math. Soc.).

[Na] M. Nagata, Polynomial rings and affine spaces, Regional Conference Series in Mathematics, 37. American Mathematical Society, Providence, R.I., 1978.

[Ve] S. Vénéreau, Automorphismes et variables de l’anneau de polynômes A[y_{1}, ..., y_{n}], Thèse de doctorat, Institut Fourier des mathématiques, Grenoble, France, 2001, http://www-fourier.ujf-grenoble.fr/ .

Department of Mathematics, University of Miami, Coral Gables, FL 33124, U.S.A.

  E-mail address: Shulim.Kaliman@math.miami.edu

Université Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St. Martin d’Hères cedex, France

  E-mail address: Mikhail.Zaidenberg@ujf-grenoble.fr