The study demonstrates conditions limiting the kernel of specific derivations in Noetherian UFDs, suggesting structural insights into their generation.
Let [Formula: see text] be a Noetherian UFD containing [Formula: see text] and [Formula: see text]. We show that, under certain conditions, the kernel of the following type of [Formula: see text]-derivations [Formula: see text] of [Formula: see text] is generated by at most [Formula: see text] elements over [Formula: see text]: a) [Formula: see text] is an extension of an [Formula: see text]-elementary derivation [Formula: see text] of [Formula: see text] such that [Formula: see text] or [Formula: see text] is a homogeneous polynomial in [Formula: see text] and [Formula: see text] with respect to the standard grading. b) [Formula: see text] is a nice [Formula: see text]-derivation satisfying [Formula: see text] and [Formula: see text] form a regular sequence or are comaximal in [Formula: see text]. c) [Formula: see text] is triangular monomial. d) [Formula: see text] is an extension of an irreducible triangular [Formula: see text]-derivation [Formula: see text] of [Formula: see text] such that [Formula: see text].
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Das et al. (2026) studied this question.
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