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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
##### Keywords:
Jacobian conjecture; derivation
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##### References:
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