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Jacobian conjecture and differential operators. (Conjecture Jacobienne et opérateurs différentiels.) (French) Zbl 0749.13003
The Jacobian conjecture asserts that an inclusion \(F:A=\mathbb{C}[x_ 1,\ldots,x_ n]\to\mathbb{C}[t_ 1,\ldots,t_ n]=B\) of complex polynomial algebras with Jacobian \(J=1\) is an equality. The assumption \(J=1\) implies that \(B\) is étale over \(A\) so that every derivation of \(A\) extends to \(B\). Let \(\delta_ i\) denote the extension of \(\partial/\partial_ i\). Then \(B\) becomes a module over the Weyl algebra \(W=\mathbb{C}[x_ 1,\ldots,x_ n,\delta_ 1,\ldots,\delta_ n]\) and hence over its Lie subalgebra \({\mathfrak gl}_ n=\text{span} x_ i\delta_ j\). The author observes that if \({\mathfrak G}\subseteq{\mathfrak gl}_ n\) is a Lie subalgebra of dimension exceeding \(n\) then \(B\) is a torsion module over the universal enveloping algebra \(U=U({\mathfrak G})\). Hence if \(B/A\) is a torsion free \(U\)-module then \(A=B\). Restricting attention to the case \(n=2\), the author studies the action of the subalgebra spanned by \(\varepsilon_ 1=x_ 1\delta_ 1\), \(\varepsilon_ 2=x_ 2\delta_ 2\) and \(\delta=x_ 1\delta_ 2\). He proves that \(B/A\) is torsion free over \(\mathbb{C}[\varepsilon_ 1+\varepsilon_ 2,\delta]\) and over \(\mathbb{C}[\varepsilon_ 1,\varepsilon_ 2]\). The latter result, surprisingly, uses results on algebraic curves with infinitely many integer points, as well as results on lacunary series.
Reviewer: A.R.Magid (Norman)
MSC:
13B10 Morphisms of commutative rings
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13N05 Modules of differentials
13B25 Polynomials over commutative rings
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