<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A$</EquationSource> <EquationSource Format="MATHML"><math> <mi>A</mi> </math></EquationSource> </InlineEquation> be an affine factorial domain over a field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$K$</EquationSource> <EquationSource Format="MATHML"><math> <mi>K</mi> </math></EquationSource> </InlineEquation> of characteristic zero endowed with an irreducible locally nilpotent derivation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation>. Assume that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> has the freeness property, its kernel is affine over <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$K$</EquationSource> <EquationSource Format="MATHML"><math> <mi>K</mi> </math></EquationSource> </InlineEquation> and its plinth ideal is generated by a power of a prime element in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\ker (\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>ker</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The main result of this paper asserts that the differential algebra <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(A,\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$K$</EquationSource> <EquationSource Format="MATHML"><math> <mi>K</mi> </math></EquationSource> </InlineEquation>-isomorphic to the coordinate ring of a generalized Danielewski variety endowed with a Jacobian-type derivation.</p>

The Freeness Property for Locally Nilpotent Derivations of Factorial Domains

期刊论文
发表日期：

2026年5月15日

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M’hammed El Kahoui,
Hatim Labtaini,
Mustapha Ouali

摘要

Let $A$ $A$ be an affine factorial domain over a field $K$ $K$ of characteristic zero endowed with an irreducible locally nilpotent derivation $\xi $ $ξ$ . Assume that $\xi $ $ξ$ has the freeness property, its kernel is affine over $K$ $K$ and its plinth ideal is generated by a power of a prime element in $\ker (\xi )$ $ker (ξ)$ . The main result of this paper asserts that the differential algebra $(A,\xi )$ $(A, ξ)$ is $K$ $K$ -isomorphic to the coordinate ring of a generalized Danielewski variety endowed with a Jacobian-type derivation.